# On the orders of vanishing elements of finite groups

Sesuai Y. Madanha  Department of Mathematics and Applied Mathematics,
University of Pretoria, Private Bag X20, Hatfield, Pretoria 0028, South Africa
<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a finite group and $p$ be a prime. Let ${\mathrm{Vo}}(G)$ denote
the set of the orders of vanishing elements, ${\mathrm{Vo}}_{p}(G)$ be the
subset of ${\mathrm{Vo}}(G)$ consisting of those orders of vanishing elements
divisible by $p$ and ${\mathrm{Vo}}_{p^{\prime}}(G)$ be the subset of
${\mathrm{Vo}}(G)$ consisting of those orders of vanishing elements not
divisible by $p$. Dolfi, Pacifi, Sanus and Spiga proved that if $a$ is not a
$p$-power for all $a\in{\mathrm{Vo}}(G)$, then $G$ has a normal Sylow
$p$-subgroup. In another article, the same authors also show that if if
${\mathrm{Vo}}_{p^{\prime}}(G)=\emptyset$, then $G$ has a normal nilpotent
$p$-complement. These results are variations of the well known Ito-Michler and
Thompson theorems. In this article we study solvable groups such that
$|{\mathrm{Vo}}_{p}(G)|=1$ and show that $P^{\prime}$ is subnormal. This is
analogous to the work of Isaacs, Moréto, Navarro and Tiep where they
considered groups with just one character degree divisible by $p$. We also
study certain finite groups $G$ such that $|{\mathrm{Vo}}_{p^{\prime}}(G)|=1$
and we prove that $G$ has a normal subgroup $L$ such that $G/L$ a normal
$p$-complement and $L$ has a normal $p$-complement. This is analogous to the
recent work of Giannelli, Rizo and Schaeffer Fry on character degrees with a
few $p^{\prime}$-character degrees. Bubboloni, Dolfi and Spiga studied finite
groups such that every vanishing element is of order $p^{m}$ for some integer
$m\geqslant 1$. As a generalization, we investigate groups such that
$\gcd(a,b)=p^{m}$ for some integer $m\geqslant 0$, for all
$a,b\in{\mathrm{Vo}}(G)$. We also study finite solvable groups whose
irreducible characters vanish only on elements of prime power order.

###### Key words and phrases:

orders of vanishing elements, $p^{\prime}$-elements, normal $p$-complements

###### 2010 Mathematics Subject Classification:

Primary 20C15

‘

##  1\. Introduction

One of the interesting problems in character theory of finite groups is
determine the structure a group using information from its character table.
Below are some results to this effect. Let $G$ be a finite group and $p$ be a
prime. Let ${\mathrm{Irr}}(G)$ denote the set of complex irreducible
characters of $G$, ${\mathrm{cd}}(G)$ the set of character degrees of $G$,
${\mathrm{cd}}_{p}(G)$ the subset of ${\mathrm{cd}}(G)$ consisting of
character degrees divisible by $p$ and ${\mathrm{cd}}_{p^{\prime}}(G)$ the
subset of ${\mathrm{cd}}(G)$ consisting of those character degrees not
divisible by $p$. A classical theorem of Thompson [17, Corollary 12.2] states
that if $|{\mathrm{cd}}_{p^{\prime}}(G)|=1$, then
$\textbf{O}^{pp^{\prime}}(G)=1$, where
$\textbf{O}^{pp^{\prime}}(G)=\textbf{O}^{p^{\prime}}(\textbf{O}^{p}(G))$. Let
$\textbf{O}^{pp^{\prime}pp^{\prime}}(G)=\textbf{O}^{pp^{\prime}}(\textbf{O}^{pp^{\prime}}(G))$.
Recently, Giannelli, Rizo and Schaeffer Fry [13] proved a variation Thompson’s
theorem by showing the following result:

######  Theorem 1.1.

[13, Theorem A] Let G be a finite group and let $p>3$ be a prime. Suppose that
$|{\mathrm{cd}}_{p^{\prime}}(G)|=2$. Then G is solvable and
$\textbf{O}^{pp^{\prime}pp^{\prime}}(G)=1$.

We are interested in studying sets of vanishing elements with some
restrictions corresponding to these results. An element $g\in G$ is a
vanishing element of $G$ if there exists $\chi\in{\mathrm{Irr}}(G)$ such that
$\chi(g)=0$. The set of all vanishing elements of G is denoted by Van(G). A
classical theorem of Burnside [17, Theorem 3.15] states that
${\mathrm{Van}}(G)$ is non-empty if there is a non-linear
$\chi\in{\mathrm{Irr}}(G)$. Let ${\mathrm{Vo}}(G)$ denote the set of the
orders of elements in ${\mathrm{Van}}(G)$, ${\mathrm{Vo}}_{p}(G)$ be the
subset of ${\mathrm{Vo}}(G)$ consisting of orders of vanishing elements
divisible by $p$ and ${\mathrm{Vo}}_{p^{\prime}}(G)$ be the subset of
${\mathrm{Vo}}(G)$ consisting of those orders of vanishing elements not
divisible by $p$.

Dolfi Pacifi, Sanus and Spiga proved the following result analogous to
Thompson’s Theorem:

######  Theorem 1.2.

[8, Corollary B] Let $G$ be a finite group and let $p$ be a prime. Suppose
that ${\mathrm{Vo}}_{p^{\prime}}(G)=\emptyset$. Then
$\textbf{O}^{pp^{\prime}}(G)=1$.

As a variation of Theorem 1.2, we study certain finite groups $G$ such that
$|{\mathrm{Vo}}_{p^{\prime}}(G)|=1$ with an extra condition and obtain our
first result.

######  Theorem A.

Let $G$ be a finite group and let $p$ and $q$ be distinct primes. Suppose that
$|{\mathrm{Vo}}_{p^{\prime}}(G)|=1$. Then G is solvable. Moreover, suppose
that one of the following holds:

  * (a)

${\mathrm{Vo}}_{p^{\prime}}(G)=\\{b\\}$ such that $b$ is divisible by at least
two primes;

  * (b)

${\mathrm{Vo}}_{p^{\prime}}(G)=\\{q^{n}\\}$ for some positive integer $n$ and
$Q^{\prime}$ is subnormal, where $Q$ is a Sylow $q$-subgroup of $G$.

Then $\textbf{O}^{pp^{\prime}pp^{\prime}}(G)=1$.

We remark that it is not clear if the hypothesis that $Q^{\prime}$ is
subnormal in $G$ is necessary. In our proof of Theorem A, we use a theorem of
Franciosi, de Giovanni, Heineken and Newell (see Lemma 2.9) which states that
if $G=AB$ is a product of an abelian group $A$ and a nilpotent group $B$, then
$A\mathbf{F}(G)$ is normal in $G$ and $G$ has Fitting height at most $3$. In
general, it is well known that for any positive integer $n$, there is group of
order $p^{a}q^{b}$ and Fitting height $n$. Hence a Fitting height of a product
of two Sylow subgroups can be arbitrarily large.

The opposite of Thompson’s theorem is the famous Ito-Michler theorem. It
states that if ${\mathrm{cd}}_{p}(G)=\emptyset$, then $G$ has a normal and
abelian Sylow $p$-subgroup. As a generalization of this, Isaacs, Moréto,
Navarro and Tiep [18, Theorem A] proved that if $|{\mathrm{cd}}_{p}(G)|=1$,
then $P^{\prime}$ is abelian and subnormal in $G$. In particular,
$P/\textbf{O}_{p}(G)$ is cyclic when $G$ is solvable. Another generalization
of the Ito-Michler theorem in terms of vanishing elements was proved by Dolfi,
Pacifi, Sanus and Spiga [8]. They showed that if all $p$-elements are non-
vanishing elements of $G$, then $G$ has a normal Sylow $p$-subgroup and that
the Sylow $p$-subgroup need not be abelian. Their result implies that if
${\mathrm{Vo}}_{p}(G)=\emptyset$, then $G$ has a normal Sylow $p$-subgroup. As
an analogue of the theorem of Isaacs, Moréto, Navarro and Tiep [18, Theorem
A(b)] for solvable groups, we consider the following for solvable groups.

######  Theorem B.

Let $G$ be a finite solvable group. If $|{\mathrm{Vo}}_{p}(G)|=1$, then
$P^{\prime}$ is subnormal in $G$. In particular, $P/\textbf{O}_{p}(G)$ is
cyclic.

We now discuss our next problem. Bubboloni, Dolfi and Spiga studied finite
groups such that every vanishing element is of order $p^{m}$ for some integer
$m>1$. They proved the following result:

######  Theorem 1.3.

[5, Corollary B] Let $G$ be a finite group and $p$ be a prime. If $a=p^{m}$
for some integer $m\geqslant 1$, for all $a\in{\mathrm{Vo}}(G)$, then one of
the following holds:

  * (a)

$G$ is a $p$-group.

  * (b)

$G/\mathbf{Z}(G)$ is a Frobenius group with a Frobenius complement of
$p$-power order and $\mathbf{Z}(G)=\textbf{O}_{p}(G)$.

Observe that groups described in Theorem 1.3 are such that
$\textbf{O}^{pp^{\prime}}(G)=1$. Consider finite groups $G$ with the following
extremal property for a fixed prime $p$:

$\gcd(a,b)=p^{m}$ for some integer $m\geqslant 0$, for all
$a,b\in{\mathrm{Vo}}(G)$.

Property (1) is a generalization of the property in Theorem 1.3. For if
$a=p^{m}$ for some integer $m\geqslant 1$, for all $a\in{\mathrm{Vo}}(G)$,
then $\gcd(a,b)=p^{k}$ for some integer $k\geqslant 1$, for all
$a,b\in{\mathrm{Vo}}(G)$. Conversely, $\mathrm{S}_{4}$ and $\mathrm{A}_{5}$
are examples of groups that satisfy property (1) for $p=2$ but do not satisfy
the property in Theorem 1.3. Property (1) is also a generalization of a
property studied in [19].

Using the classification of finite simple groups, we show that if $G$
satisfies property (1) for an odd prime $p>7$, then
$\textbf{O}^{pp^{\prime}pp^{\prime}}(G)=1$:

######  Theorem C.

Let $G$ be a finite group and $p>7$ be a prime. If $G$ satisfies property (1)
for $p$, then $G$ is solvable and $\textbf{O}^{pp^{\prime}pp^{\prime}}(G)=1$.

The condition that $p>7$ is necessary in Theorem C : $\mathrm{A}_{7}$ shows
the theorem fails for any $p\leqslant 7$.

Zhang, Li and Shao [26] studied finite non-solvable groups whose irreducible
characters vanish only on elements of prime power order. We investigate finite
solvable groups whose irreducible characters vanish only on elements of prime
power order. This is another generalization of Theorem 1.3. We show that
either these groups have vanishing elements which are of $p$-power order or
they are groups of order $p^{a}q^{b}$ for some primes $p$ and $q$.

######  Theorem D.

Let $G$ be a finite non-abelian solvable group and $p,q$ be distinct primes.
If every irreducible character of $G$ vanishes on elements of prime power
order only, then one of the following holds.

  * (i)

$G$ is a $p$-group;

  * (ii)

$G/\textbf{Z}(G)$ is a Frobenius group with a Frobenius complement of
$p$-power order and $\textbf{Z}(G)=\textbf{O}_{p}(G)$;

  * (iii)

$G$ is a Frobenius group with a Frobenius complement of $p$-power order and a
kernel of $q$-power order, that is, $\pi(G)=\\{p,q\\}$;

  * (iv)

$G$ is a nearly $2$-Frobenius group and $\pi(G)=\\{p,q\\}$.

Higman [16] described finite groups in which every element is of prime power
order. In particular, he proved that the solvable groups with this property
have orders that are divisible by at most two distinct primes. Hence Theorem D
generalizes Higman’s result for solvable groups. Note that groups in Theorem
D(ii) have orders that may be divisible by more than two distinct primes (see
[5, Proposition 5.4]).

##  2\. Preliminaries

In this section we shall list some properties of vanishing elements.

######  Lemma 2.1.

[22, Lemma 2] Let $G$ be a finite solvable group. Suppose $M,N$ are normal
subgroups of $G$.

  * (a)

If $M\setminus N$ is a conjugacy class of $G$ and $\gcd(|M{:}N|,|N|)=1$, then
$M$ is a Frobenius group with kernel $N$ and prime order complement.

  * (b)

If $G\setminus N$ is a conjugacy class of $G$, then $G$ is a Frobenius group
with an abelian kernel and complement of order two.

The existence of $p$-defect zero characters is guaranteed in finite simple
groups $G$ for all primes $p\geqslant 5$ dividing $|G|$ as the following
result shows:

######  Lemma 2.2.

[14, Corollary 2.2] Let $G$ be a non-abelian finite simple group and $p$ be a
prime. If $G$ is a finite group of Lie type, or if $p\geqslant 5$, then there
exists $\chi\in{\mathrm{Irr}}(G)$ of $p$-defect zero.

######  Lemma 2.3.

[3, Lemma 2.2] Let $G$ be a finite group, $N$ a normal subgroup of $G$ and $p$
a prime. If $N$ has an irreducible character of $p$-defect zero, then every
element of $N$ of order divisible by $p$ is a vanishing element in $G$.

######  Lemma 2.4.

[2, Lemma 5] Let $G$ be a finite group, and $N=S_{1}\times\cdots\times S_{k}$
a minimal normal subgroup of $G$, where every $S_{i}$ is isomorphic to a non-
abelian simple group $S$. If $\theta\in{\mathrm{Irr}}(S)$ extends to
${\mathrm{Aut}}(S)$, then
$\varphi=\theta\times\cdots\times\theta\in{\mathrm{Irr}}(N)$ extends to $G$.

######  Lemma 2.5.

[20, Theorem 1.1] Suppose that $N$ is a minimal normal non-abelian subgroup of
a finite group $G$. Then there exists an irreducible character $\theta$ of $N$
such that $\theta$ is extendible to $G$ with $\theta(1)\geqslant 5$.

Given a finite set of positive integers $Y,$ the prime graph $\Pi(Y)$ is
defined as the undirected graph whose vertices are the primes $p$ such that
there exists an element of $Y$ divisible by $p$, and two distinct vertices
$p,q$ are adjacent if and only if there exists an element of $Y$ divisible by
$pq$. The vanishing prime graph of $G$, denoted by $\Gamma(G)$, is the prime
graph $\Pi(\mathrm{Vo}(G))$. For a group $G$, let $\omega(G)$ be the set of
orders of elements of $G$. We shall state a result on non-solvable groups with
disconnected vanishing prime graphs. Let $n(\mathcal{G})$ be the number of
connected components of the graph $\mathcal{G}$.

######  Theorem 2.6.

[9, Theorem B] Let $G$ be a finite non-solvable group. If $\Gamma(G)$ is
disconnected, then $G$ has a unique non-abelian composition factor $S$, and
$n(\Gamma(G))\leqslant n(\Pi(\omega(S)))$ unless $G$ is isomorphic to
$\mathrm{A}_{7}$.

We shall also state a result on solvable groups with disconnected vanishing
prime graphs. We first recall two definitions:

A group $G$ is said to be a _$2$ -Frobenius group_ if there exists two normal
subgroups $F$ and $L$ of $G$ such that $G/F$ is a Frobenius group with kernel
$L/F$ and $L$ is a Frobenius group with kernel $F$.

A group $G$ is said to be a _nearly $2$-Frobenius group_ if there exist two
normal subgroups $F$ and $L$ of $G$ with the following properties:
$F=F_{1}\times F_{2}$ is nilpotent, where $F_{1}$ and $F_{2}$ are normal
subgroups of $G$. Furthermore, $G/F$ is a Frobenius group with kernel $L/F$,
$G/F_{1}$ is a Frobenius group with kernel $L/F_{1}$, and $G/F_{2}$ is a
$2$-Frobenius group.

######  Theorem 2.7.

[10, Theorem A] Let $G$ be a finite solvable group. Then $\Gamma(G)$ has at
most two connected components. Moreover, if $\Gamma(G)$ is disconnected, then
$G$ is either a Frobenius group or a nearly $2$-Frobenius group.

The following is a classification of Frobenius complements.

######  Theorem 2.8.

[4, Theorem 1.4] Let $G$ be a Frobenius group with Frobenius complement $M$.
Then $M$ has a normal subgroup $N$ such that all Sylow subgroups of $N$ are
cyclic and one of the following holds:

  * (a)

$M/N\cong 1$;

  * (b)

$M/N\cong\mathrm{V}_{4}$, the Sylow $2$-subgroup of the alternating group
$\mathrm{A}_{4}$;

  * (c)

$M/N\cong\mathrm{A}_{4}$;

  * (d)

$M/N\cong\mathrm{S}_{4}$;

  * (e)

$M/N\cong\mathrm{A}_{5}$;

  * (f)

$M/N\cong\mathrm{S}_{5}$.

######  Lemma 2.9.

[12, Lemma 3] Let the finite group $G=AB$ be the product of an abelian
subgroup $A$ and a nilpotent subgroup $B$. Then $A\mathbf{F}(G)$ is a normal
subgroup of $G$. In particular, the Fitting height of $G$ is at most $3$.

##  3\. A reduction theorem

We shall prove a reduction theorem in this section. We shall first prove the
proposition below. Let
$\mathfrak{S}=\\{\mathrm{A}_{5},\mathrm{A}_{6},\mathrm{A}_{7},{\mathrm{PSL}}_{2}(7),{\mathrm{PSL}}_{2}(8),{\mathrm{PSL}}_{3}(4),^{2}\\!\mathrm{B}_{2}(8)\\}$.

######  Proposition 3.1.

Let $N$ be a non-abelian minimal normal subgroup of $G$ such that
$N\notin\mathfrak{S}$. Then one of the following holds

  * (a)

There are distinct primes $p_{1},p_{2}\mid|N|$ and vanishing elements $x,y,z$
of $G$ contained in $N$ such that ${\mathrm{ord}}(x)=p_{1}$,
${\mathrm{ord}}(y)=p_{2}$ and $p_{1}p_{2}\mid{\mathrm{ord}}(z)$.

  * (b)

There are distinct primes $p_{1},p_{2}\mid|N|$ and vanishing elements
$x_{1},x_{2},y_{1},y_{2}$ of $G$ such that
${\mathrm{ord}}(x_{i})\not={\mathrm{ord}}(y_{i})$,
$p_{i}\mid{\mathrm{ord}}(x_{i})$ and $p_{i}\mid{\mathrm{ord}}(y_{i})$ for
$i=1,2$.

  * (c)

There is a prime $p\neq 3$ such that $p\mid|N|$ and vanishing elements $x,y,z$
of $G$ such that ${\mathrm{ord}}(x)=3$, ${\mathrm{ord}}(y)=3p$ and
${\mathrm{ord}}(z)=p^{2}$.

  * (d)

There is an odd prime $p\mid|N|$ and vanishing elements $x,y,z$ of $G$ such
that ${\mathrm{ord}}(x)=p$, ${\mathrm{ord}}(y)=2p$ and ${\mathrm{ord}}(z)=4p$.

###### Proof.

Let $N=S_{1}\times S_{2}\times\cdots\times S_{k}$, where $S_{i}\cong S$, $S$
is a simple group. Suppose that $k\geqslant 2$. Using [17, Theorem 3.10],
$\pi(|S|)\geqslant 3$. Suppose that $\pi(|S|)\geqslant 4$. Then there are
distinct primes $p_{1},p_{2}\geqslant 5$ such that $p_{i}\mid|N|$ and so $N$
has an irreducible character of $p_{i}$-defect zero for $i=1,2$ by Lemma 2.2.
Let $x_{i}\in S_{i}$ be a $p_{i}$-element for $i=1,2$. Note that $x_{i}$ is a
vanishing element of $G$ by Lemma 2.3. The element $x_{1}x_{2}$ is a vanishing
element and ${\mathrm{ord}}(x_{1}x_{2})$ is divisible by $p_{1}p_{2}$. This is
case (a).

We may assume that $|\pi(G)|=3$. Using a result of Herzog in [15], we have
that
$S\in\\{{\mathrm{PSL}}_{2}(5),{\mathrm{PSL}}_{2}(7),{\mathrm{PSL}}_{2}(8),{\mathrm{PSL}}_{2}(9),{\mathrm{PSL}}_{2}(17),{\mathrm{PSL}}_{3}(3),\mathrm{PSU}_{3}(3),\mathrm{PSU}_{4}(2)\\}$.
In particular, $S$ is a simple group of Lie type. By Lemma 2.2, $S$ and hence
by extension $N$ has an irreducible character of $p$-defect zero for all
$p\mid|N|$. Let $x_{1}\in S_{1}$ be a $2$-element and $x_{2}\in S_{2}$ be a
$p$-element for some odd prime $p$. Now $x_{1}$, $x_{2}$ and $x_{1}x_{2}$ are
vanishing elements and the order of $x_{1}x_{2}$ is divisible by $2p$. This is
case (a).

We may assume that $N$ is a simple group. Let $N$ be a sporadic simple group
or ${}^{2}\mathrm{F}_{4}(2)^{\prime}$. Table 3.1 below contains an irreducible
characters $\theta_{1}$ and $\theta_{2}$ of $N$ of $p$-defect zero and
$q$-defect zero for some primes $p$ and $q$, respectively, and three elements
of distinct orders divisible by $p$, $q$ and $pq$, respectively. This is case
(a) and it follows from Lemma 2.3. We shall use the character tables and
notation in the Atlas [7].

Table 3.1  $N$ $\theta_{1}(1)$ $\theta_{2}(1)$
${\mathrm{ord}}(v\mathcal{C}_{1})$ ${\mathrm{ord}}(v\mathcal{C}_{2})$
${\mathrm{ord}}(v\mathcal{C}_{3})$ $M_{11}$ $\chi_{9}(1)=45$ $\chi_{6}(1)=16$
$3A$ $2A$ $6A$ $M_{23}$ $\chi_{3}(1)=45$ $\chi_{12}=896$ $3A$ $2A$ $6A$
$M^{c}L$ $\chi_{9}(1)=1750$ $\chi_{7}=896$ $5A$ $2A$ $10A$ $Co_{2}$
$\chi_{5}(1)=1771$ $\chi_{54}=1835008$ $7A$ $2A$ $14A$ $He$ $\chi_{9}(1)=1275$
$\chi_{54}(1)=21504$ $5A$ $2A$ $10A$ $Fi_{22}$ $\chi_{4}(1)=1001$
$\chi_{56}(1)=1441792$ $7A$ $2A$ $14A$ $Fi_{23}$ $\chi_{8}(1)=106743$
$\chi_{94}(1)=504627200$ $7A$ $2A$ $14A$ $Fi_{24}^{\prime}$
$\chi_{4}(1)=249458$ $\chi_{97}(1)=197813862400$ $11A$ $2A$ $22A$ $HN$
$\chi_{2}(1)=133$ $\chi_{46}(1)=3424256$ $7A$ $2A$ $14A$ $Th$
$\chi_{6}(1)=30628$ $\chi_{22}(1)=4096000$ $13A$ $3A$ $39A$ $M$ $\chi_{9}(1)$
$\chi_{102}(1)$ $17A$ $2A$ $34A$ $J_{1}$ $\chi_{9}(1)=120$ $\chi_{2}(1)=56$
$3A$ $2A$ $6A$ $O^{\prime}N$ $\chi_{8}(1)=32395$ $\chi_{23}(1)=175616$ $5A$
$2A$ $10A$ $J_{3}$ $\chi_{2}(1)=85$ $\chi_{14}(1)=1920$ $5A$ $2A$ $10A$ $Ly$
$\chi_{5}(1)=48174$ $\chi_{7}(1)=120064$ $7A$ $2A$ $14A$ $J_{4}$
$\chi_{9}(1)=1187145$ $\chi_{53}(1)=1981808640$ $5A$ $2A$ $10A$
${}^{2}\mathrm{F}_{4}(2)^{\prime}$ $\chi_{8}(1)=325$ $\chi_{21}=2048$ $5A$
$2A$ $10A$

For the sporadic simple groups in Table 3.2, we have two irreducible
characters $\theta_{1}$ and $\theta_{2}$ of $N$ of $p_{1}$-defect zero and
$p_{2}$-defect zero, respectively. We also have two $p_{i}$-elements of
distinct orders for each $\theta_{i}$ on which $\theta_{i}$ vanishes. This is
case (b). For the case when $N\cong Suz$, we have that $N$ has an irreducible
character of $5$-defect zero and elements of orders $5,10$ and $20$. In the
case when $N\cong HS$, $N$ has an irreducible character of $3$-defect zero and
elements of orders $3,6$ and $12$. Both of these cases above satisfy case (d).

Table 3.2  $N$ $\theta_{1}(1)$ ${\mathrm{ord}}(v\mathcal{C}_{1})$
${\mathrm{ord}}(v\mathcal{C}_{2})$ $\theta_{2}(1)$
${\mathrm{ord}}(v\mathcal{C}_{3})$ ${\mathrm{ord}}(v\mathcal{C}_{4})$ $M_{12}$
$\chi_{8}(1)=55$ $5A$ $10A$ $\chi_{7}(1)=54$ $3A$ $6A$ $M_{24}$
$\chi_{3}(1)=45$ $5A$ $10A$ $\chi_{26}(1)=10395$ $3A$ $6A$ $J_{2}$
$\chi_{18}(1)=225$ $5A$ $10A$ $\chi_{14}(1)=189$ $3A$ $6A$ $Co_{3}$
$\chi_{6}(1)=896$ $7A$ $14A$ $\chi_{23}(1)=31625$ $5A$ $10A$ $Co_{1}$
$\chi_{4}(1)=1771$ $11A$ $22A$ $\chi_{3}(1)=299$ $13A$ $26A$ $B$
$\chi_{11}(1)=3214743741$ $11A$ $22A$ $\chi_{8}(1)=347643114$ $13A$ $26A$ $Ru$
$\chi_{2}(1)=378$ $3A$ $6A$ $\chi_{2}(1)=378$ $7A$ $14A$

We are left with the case when $N\cong M_{22}$. There exists
$\theta_{1}=\chi_{10}$, $\theta_{2}=\chi_{3}$ which are extendible to $G$ by
Lemma 2.4. Note that $\chi_{3}(3A)=\chi_{3}(6A)=0$ and $\chi_{10}(4A)=0$ and
hence this is case (c).

Suppose that $N$ is an alternating group $\mathrm{A}_{n}$, $n\geqslant 5$.
Since $N\notin\mathfrak{S}$, $n\geqslant 8$. Suppose $N\cong\mathrm{A}_{8}$ or
$\mathrm{A}_{9}$. Then $N$ has two irreducible characters $\chi_{15}$ and
$\chi_{7}$ of $3$-defect zero and $5$-defect zero, respectively. The result
follows because $N$ has elements of orders $3$, $5$ and $15$. This is case
(a).

Suppose $N\cong\mathrm{A}_{10}$. By Lemma 2.2, $N$ has irreducible characters
$\theta_{1}$ and $\theta_{2}$, of $5$-defect zero and $7$-defect zero,
respectively. Since $N$ has elements of orders $5$, $10$, $7$ and $21$, the
result follows using Lemma 2.3. This is case (b).

Suppose that $N$ is an alternating group $\mathrm{A}_{n}$, $n\geqslant 11$. By
Lemma 2.2, $N$ has an irreducible character is of $5$-defect zero. Note that
$N$ has three elements $(12345)$, $(12345)(6789)(10~{}11)$ and
$(12345)(67)(89)$ of orders $5$, $10$ and $20$, respectively. Using Lemma 2.3,
we have three elements are vanishing elements of $G$. This is case (d).

We may assume that $N$ is simple group of Lie type. Suppose that $N$ has an
element of order $2s$, $s$ an odd prime. Then $N$ has irreducible characters
(not necessarily distinct) $\theta_{1}$ and $\theta_{2}$ of $2$-defect zero
and of $s$-defect zero by Lemma 2.2. Since $N$ has elements of orders $2$, $s$
and $2s$, the result follows since these two elements are vanishing elements
by Lemma 2.3. This is case (a). We may assume that $N$ has no element of order
$2s$, $s$ an odd prime. Then the centralizer of each involution contained in
$N$ is a $2$-group. It follows from [24, III, Theorem 5] that $N$ is
isomorphic to one of the following: ${\mathrm{PSL}}_{2}(r)$, where $r$ is a
Fermat or Mersenne prime; ${\mathrm{PSL}}_{2}(9)$;
$N\cong{\mathrm{PSL}}_{3}(4)$; $N\cong^{2}\\!\\!\mathrm{B}_{2}(2^{2n+1})$,
$n\geqslant 1$.

Since $N\notin\mathfrak{S}$, we may assume that $N\cong{\mathrm{PSL}}_{2}(r)$,
where $r$ is a Fermat or Mersenne prime and $r\geqslant 17$ or
$N\cong^{2}\\!\\!\mathrm{B}_{2}(2^{2n+1})$, $n\geqslant 2$. Suppose that
$N\cong{\mathrm{PSL}}_{2}(r)$, $r\geqslant 17$. Suppose that
$N\cong{\mathrm{PSL}}_{2}(17)$. Then $G$ has vanishing elements of orders
$2,4,3$ and $9$. If $N\cong{\mathrm{PSL}}_{2}(31)$, then $G$ has vanishing
elements of orders $3,5$ and $15$. The result follows. Suppose that
$N\cong{\mathrm{PSL}}_{2}(r)$, where $r\geqslant 127$ is a Fermat or Mersenne
prime. The character tables of ${\mathrm{PSL}}_{2}(q)$ are well known (see for
example [11, Chapter 38]). In particular $N$ has vanishing elements orders
which are factors of $(r+1)/2$ and $(r-1)/2$. Since $r$ is a Fermat or
Mersenne prime either $r-1=2^{n}$ or $r+1=2^{n}$ for positive integer $n$.
This means that $N$ has elements of orders $2$ and $4$. Hence either $r+1$ or
$r-1$ is divisible by $3$. Since $r\geqslant 157$, we have that $r\pm
1=3^{k}$, $k\geqslant 5$ or $r\pm 1$ is divisible by at least distinct primes.
Hence $N$ has vanishing elements of orders $2,4,3,9$ or $3,q,3q$ for some
prime $q\neq 3$. In other words, this is case (b) or case (a).

Suppose that $N\cong^{2}\\!\\!\mathrm{B}_{2}(2^{2n+1})$ with $n\geqslant 2$.
In this case, $N$ has cyclic Hall subgroups $H_{1}$, $H_{2}$ and $H_{3}$ with

$|H_{1}|=2^{2n+1}-2^{n+1}+1=r_{1}$, $|H_{2}|=2^{2n+1}+2^{n+1}+1=r_{2}$ and
$|H_{3}|=2^{2n+1}-1=r_{3}$.

Orders of elements of $N$ consist of factors of $r_{1}$, $r_{2}$ and $r_{3}$.
Note that all the non-trivial elements of $N$ are vanishing elements of $G$.
If $r_{i}$ is not prime for some $i=1,2,3$, then the result follows since $N$
has elements of order $2$ and $4$. This is case (b). Hence we may assume that
$r_{1}$, $r_{2}$ and $r_{3}$ are prime. This means $|\pi(N)|=4$. Using [6], we
have that $N\cong^{2}\\!\\!\mathrm{B}_{2}(32)$. Now $N$ has elements of orders
$2$, $4$, $5$ and $25$ which are vanishing elements of $G$ and the result
follows. This is case (b) and this concludes our argument. ∎

We are now ready to prove our reduction theorem below:

######  Theorem 3.2.

(Reduction Theorem) Let $G$ be a finite group and let $p$ be a prime. If $G$
satisfies property (1) for $p$, then there exists a solvable normal subgroup
$N$ of $G$ such that one of the following holds:

  * (a)

$G$ is solvable;

  * (b)

$p=2,3$ or $5$, $G/N\cong\mathrm{A}_{5}$ and one of the following holds:

    * (i)

$N/\textbf{O}^{2}(N)$ is an elementary abelian group;

    * (ii)

$N/\textbf{O}_{\pi^{\prime}}(G)$ is a $\pi$-group with $\pi=\\{2,3,5\\}$, that
is, $\textbf{O}_{\pi^{\prime}}(G)$ is a normal Hall $\pi^{\prime}$-subgroup of
$G$;

  * (c)

$p=2$ and $G/N\in\\{\mathrm{A}_{6},\mathrm{A}_{6}{:}2_{3}\\}$;

  * (d)

$p=2,3,5$ or $7$ and $G\cong\mathrm{A}_{7}$.

  * (e)

$p=2$ and $G/N\cong{\mathrm{PSL}}_{2}(7)$;

  * (f)

$p=3$ and $G/N\cong{\mathrm{PSL}}_{2}(8)$;

  * (g)

$p=2$ and $G/N\cong{\mathrm{PSL}}_{3}(4)$;

  * (h)

$p=2$ and $G/N\cong^{2}\\!\\!\mathrm{B}_{2}(8)$.

###### Proof.

Suppose that $G$ has no composition factor isomorphic to $S\in\mathfrak{S}$.
We prove that $G$ is solvable using induction on $|G|$. Let $N$ be a non-
trivial normal subgroup of $G$. Then $G/N$ has no composition factor
isomorphic to $S$ and satisfies property (1) and hence $G/N$ is a solvable
group. If $N_{1}$ and $N_{2}$ are two minimal normal subgroups of $G$, then
$G/N_{1}$ and $G/N_{2}$ are solvable. Hence $G$ is solvable since the class of
finite solvable groups is a formation. We may assume that G has a unique non-
abelian minimal normal subgroup $N$. Then by Proposition 3.1, $G$ does not
satisfy property (1) for any prime $p$.

We may assume that $G$ has a composition factor isomorphic to
$S\in\mathfrak{S}$. Since $G$ satisfies property (1), $\Gamma(G)$ is
disconnected. By Theorem 2.6, $G$ has a unique non-abelian composition factor.
Letting $N$ to be the solvable radical of $G$, we have that $G/N$ is an almost
simple group and there exists a normal subgroup $M$ of $G$ such that $M/N$ is
isomorphic to $S\in\mathfrak{S}$.

Suppose that $M=G$ and $G/N\cong\mathrm{A}_{5}$. Then $p=2,3$ or $5$. By [21,
Lemma 2.2], $G$ has a $2$-element, a $3$-element and a $5$-element which are
vanishing elements. Let $\pi=\\{2,3,5\\}$. Suppose that there exist an $r$
element, $r\notin\pi$ prime, which is a vanishing element. Then $r$ is an
isolated vertex in the vanishing prime graph of $G$ by property (1). In other
words, $n(\Gamma(G))\geqslant 4>3=n(\Gamma(\mathrm{A}_{5}))$ by Theorem 2.6, a
contradiction. Hence if $G$ has another vanishing element other than
$q$-elements, $q\in\pi$, then it has to be a $q$-singular element $g$ such
that some prime $r\notin\pi$ divides ${\mathrm{ord}}(g)$, using property (1).
This means that $G$ has a normal Sylow $r$-subgroup. Therefore either
${\mathrm{Van}}(G)$ consists of $2$-elements, $3$-elements and $5$-elements or
$G$ has an additional vanishing element whose order is divisible by $qr$,
$q\in\pi$ and $r\notin\pi$. The first case is (b)(i) by [26, Theorem A].

For the latter case we have that $G$ has a normal Hall
$\pi^{\prime}$-subgroup. This is case (b)(ii) of the theorem.

Suppose that $M<G$ and $G/N\cong\mathrm{S}_{5}$. Then there exist elements of
orders $2$, $3$ and $6$ which are vanishing elements of $G/N$. Hence $G/N$
does not satisfy property (1) and the result follows.

If $M/N\cong\mathrm{A}_{6}$, then $M/N$ has an irreducible character of
$2$-defect zero and so vanishing elements of orders $2$ and $4$ by Lemma 2.3.
This means that $G$ does not satisfy property (1) for an odd prime $p$.
Suppose that $p=2$. Then checking the character tables in the Atlas [7], for
an almost simple group $G/N$ such that $|G/M|\leqslant 2$, our result follows
except for $\mathrm{A}_{6}$ and $\mathrm{A}_{6}{:}2_{3}$. For $G$ such that
$|G/M|=4$, we have the character table from GAP [25] and can conclude that $G$
does not satisfy property (1). Hence (c) follows.

Suppose that $M=G$ and $G/N\cong\mathrm{A}_{7}$. Then $G/N$ has vanishing
elements of orders $3$, $4$, $5$ and $7$. So $\Gamma(G/N)$ has four
components. Suppose $N_{1}$ is a normal subgroup such that $N/N_{1}$ is a
chief factor of $G$. In particular, $N/N_{1}$ is a $r$-group for some prime
$r$. By [10, Lemma 5.1], $\Gamma(G/N_{1})$ has at most two connected
components. This means that $G/N_{1}$ does not satisfy property (1). We have
that $G$ does not satisfy property (1). Hence $p=2,3,5$ or $7$, $N=1$ and this
is case (d) of the theorem.

Suppose that $M<G$ and $G/N\cong\mathrm{S}_{7}$. Then there exist elements of
orders $2,3$ and $6$ which are vanishing elements of $G/N$ and the result
follows.

Now if
$M/N\in\\{{\mathrm{PSL}}_{2}(7),{\mathrm{PSL}}_{3}(4),^{2}\\!\mathrm{B}_{2}(8)\\}$,
then $M/N$ has an irreducible character of $2$-defect zero and elements of
orders $2$ and $4$ and so $G/N$ does not satisfy property (1) for an odd prime
$p$. We may assume that $p=2$.

Suppose that $M/N\cong{\mathrm{PSL}}_{2}(7)$ or
$M/N\cong^{2}\\!\\!\mathrm{B}_{2}(8)$. Then by consulting character tables in
the Atlas [7], our result follows. Suppose $M/N\cong{\mathrm{PSL}}_{3}(4)$.
Then for an almost simple $G$ such that $|G/M|\leqslant 6$, our result follows
by checking the explicit character tables in the Atlas [7]. For the case when
$|G/M|=12$, we obtain the character table from GAP [25] and our result
follows.

Suppose that $G/N\cong{\mathrm{PSL}}_{2}(8)$. Then $p=3$ and $G$ satisfies
property (1). If $G/N\cong{\mathrm{PSL}}_{2}(8){:}3$, then $G/N$ has vanishing
elements of orders $2,3$ and $6$ and the result follows. ∎

We prove the first part of Theorem A below. First we prove a lemma.

######  Lemma 3.3.

Let $N$ be a non-abelian minimal normal subgroup of $G$. Suppose that $p$ is a
prime such that $p\mid|N|$. Then there exists a vanishing element of $G$ that
is a $p$-element.

###### Proof.

Since $N$ is non-abelian, $N=S_{1}\times S_{2}\times\cdots\times S_{k}$, where
$S_{i}\cong S$, and $S$ is non-abelian simple group. Let $S$ be a finite group
of Lie type or $p\geqslant 5$. By Lemma 2.2, $S$ has an irreducible character
$\theta$ of $p$-defect zero. Let $x$ be a $p$-element of $S$. Then
$\psi=\theta\times\theta\times\cdots\times\theta$ is an irreducible character
of $N$ of $p$-defect zero. Hence $g=x\times x\times\cdots\times x\in N$ is a
$p$-element and by Lemma 2.3, $g$ is a vanishing element of $G$.

Suppose that $p\in\\{2,3\\}$ and $S$ is a sporadic simple group or an
alternating group $\mathrm{A}_{n}$, $n\geqslant 7$. By [8, Lemma 2.3] and [8,
Proposition 2.4], there exists an irreducible character $\theta$ of $S$ that
extends to ${\mathrm{Aut}}(S)$ and $\theta(x)=0$ for some $p$-element $x\in
S$. Let $g=x\times x\times\cdots\times x\in N$ and note that $g$ is a
$p$-element. By [8, Proposition 2.2],
$\theta\times\theta\times\cdots\times\theta\in{\mathrm{Irr}}(N)$ extends to
some $\chi\in{\mathrm{Irr}}(G)$. Hence $\chi(g)=(\theta(x))^{k}=0$ and this
concludes our proof. ∎

######  Theorem 3.4.

Let $G$ be a finite group and let $p$ be a prime. Suppose that
$|{\mathrm{Vo}}_{p^{\prime}}(G)|=1$. Then $G$ is solvable.

###### Proof.

We prove that $G$ is solvable using induction on $|G|$. Let $N$ be a non-
trivial normal subgroup of $G$. Then
$|{\mathrm{Vo}}_{p^{\prime}}(G/N)|\leqslant 1$ and hence $G/N$ is a solvable
group. If $N_{1}$ and $N_{2}$ are two minimal normal subgroups of $G$, then
both $G/N_{1}$ and $G/N_{2}$ are solvable. Hence $G$ is solvable. We may
assume that $G$ has a unique non-abelian minimal normal subgroup $N$. Note
that $|\pi(N)|\geqslant 3$ by [17, Theorem 3.10]. Using Lemma 3.3, we have
that $|{\mathrm{Vo}}_{p^{\prime}}(G)|\geqslant 2$. Hence the result follows. ∎

##  4\. Solvable groups

In this section we shall restate and prove our main theorems. We start with
Theorem D.

###### Proof of Theorem D.

Suppose that $\Gamma(G)$ has one component. Then all vanishing elements are
$p$-elements for some $p$. By [5, Theorem A], $G$ is either a $p$-group or
$G/\textbf{Z}(G)$ is a Frobenius group with a Frobenius complement of
$p$-power order and $\textbf{Z}(G)=\textbf{O}_{p}(G)$. This is (a) and (b).

Suppose $\Gamma(G)$ is disconnected. By Theorem 2.7, $\Gamma(G)$ has at
exactly two connected components. This means that $G$ has vanishing elements
which are $p$-elements and $q$-elements for some distinct primes $p$ and $q$.
Note that $G$ is a Frobenius group or a nearly $2$-Frobenius group.

Suppose that $G$ is a Frobenius group. Let the Frobenius complement be of
$p$-power order for some prime $p$. Suppose that the order of the kernel $K$
is divisible by at least two distinct primes. There exists a $q$-element which
is a vanishing element of $G$ contained in $K$. Using [10, Proposition 2.1],
there exists $g$ which is a vanishing element such that
$\pi({\mathrm{ord}}(g))=\pi(|K|)$, a contradiction. Therefore $|K|$ is of
$q$-power order as expected in (c). If the Frobenius complement is divisible
by two distinct primes, $p$ and $q$, then there exists a prime $r$ that
divides $|K|$. Using [10, Proposition 3.2], there exists a vanishing element
$g$ such that ${\mathrm{ord}}(g)$ is either divisible by $pr$ or $qr$, a
contradiction.

Suppose that $G$ is a nearly $2$-Frobenius group. Then there exist two normal
subgroups $F$ and $L$ of $G$ with the following properties: $F=F_{1}\times
F_{2}$ is nilpotent, where $F_{1}$ and $F_{2}$ are normal subgroups of $G$.
Furthermore, $G/F$ is a Frobenius group with kernel $L/F$, $G/F_{1}$ is a
Frobenius group with kernel $L/F_{1}$, and $G/F_{2}$ is a $2$-Frobenius group.
Since $G/F_{2}$ is a $2$-Frobenius group and $G/F$ is a Frobenius group with
kernel $L/F$, it follows that $L/F_{2}$ is a Frobenius group with kernel
$F/F_{2}$. By the argument above, $|G/L|$ is a $p$-power and $|L/F_{1}|$ is a
$q$-power. If $|F|$ is divisible by three distinct primes, then there exists a
prime $r$ different from $p$ and $q$. Using [10, Proposition 3.2], there
exists a vanishing element $g$ such that ${\mathrm{ord}}(g)$ is either
divisible by $pr$ or $qr$, a contradiction. Hence $|F|$ is divisible by at
most two primes $p$ and $q$. The result then follows. ∎

We note here that the converse of this theorem is not true (see [5, Example
2]). The following example shows a group which satisfies properties in part
(iv) of the theorem.

###  4.1. Example

Let
$G=((\mathrm{C}_{2}\times\mathrm{C}_{2})\rtimes\mathrm{C}_{9})\rtimes\mathrm{C}_{2}$,
$L=(\mathrm{C}_{2}\times\mathrm{C}_{2})\rtimes\mathrm{C}_{9}$,
$F_{1}=\mathrm{C}_{2}\times\mathrm{C}_{2}$, $F_{2}=\mathrm{C}_{3}$ and
$F=(\mathrm{C}_{2}\times\mathrm{C}_{2})\times\mathrm{C}_{3}$. Then
$G/F\cong\mathrm{S}_{3}$ and $G/F_{1}\cong\mathrm{D}_{18}$, a Frobenius group
of with kernel $L/F_{1}\cong\mathrm{C}_{9}$. Also
$G/F_{2}\cong\mathrm{S}_{4}$, a $2$-Frobenius group. In other words, $G$ is a
nearly $2$-Frobenius group and $\mathrm{Vo}(G)=\\{2,4,9\\}$ using GAP [25].

######  Theorem 4.1.

Let $G$ be a finite solvable group and $p$ be a prime. If $G$ satisfies
property (1) for $p$, then $\textbf{O}^{pp^{\prime}pp^{\prime}}(G)=1$.

###### Proof.

We first consider when $\Gamma(G)$ is connected. Suppose that $\Gamma(G)$
consists of a single vertex. If the vertex is $q$, then by Theorem 1.3, $G$ is
a $q$-group or $G/\mathbf{Z}(G)$ is a Frobenius group with a Frobenius
complement of $q$-power order and $\textbf{Z}(G)=\textbf{O}_{q}(G)$. If $p=q$,
then $\textbf{O}^{pp^{\prime}}(G)=1$. If $p\not=q$, then $G$ has a normal
Sylow $p$-subgroup. Hence $\textbf{O}^{p}(\textbf{O}^{pp^{\prime}}(G))=1$.

Suppose that $\Gamma(G)$ has at least two vertices. Then $p\mid a$ for all
$a\in{\mathrm{Vo}}(G)$. Using Theorem 1.2, $\textbf{O}^{pp^{\prime}}(G)=1$.

Suppose that $\Gamma(G)$ is disconnected. By Theorem 2.7, $\Gamma(G)$ has two
connected components. Since $G$ satisfies property (1) for $p$, there exists
exactly one $b\in{\mathrm{Vo}}(G)$ such that $p\nmid b$. If $b$ is divisible
by two distinct primes $p_{1},p_{2}$, then $G$ has a normal Sylow
$p_{i}$-subgroup, $i=1,2$ by [8, Theorem A] since G has no $p_{i}$-elements as
vanishing elements. Hence $G$ has a normal Hall $p^{\prime}$-subgroup $H$ and
$H$ is a normal nilpotent $p$-complement of $G$.

We may assume that $b\in{\mathrm{Vo}}(G)$ and $b$ is a $q$-power for the rest
of the proof. If $a=p^{m}$ for all $a\in{\mathrm{Vo}}(G)\setminus\\{b\\}$,
then by Theorem D, $G$ is a Frobenius group with a Frobenius complement of
$r$-power order and a kernel of $s$-power order or $G$ is a nearly
$2$-Frobenius group and $\pi(G)=\\{r,s\\}$. Assume that $G$ is a Frobenius
group with a Frobenius complement of $r$-power order and a kernel $K$ of
$s$-power order. If $r=p$, then the result follows. If $s=p$, then
$\textbf{O}^{p}(G)=G$ and $\textbf{O}^{p^{\prime}}(G)=K$. Hence
$\textbf{O}^{p}(K)=1$ and the result follows. Assume that $G$ is a nearly
$2$-Frobenius group and $\pi(G)=\\{r,s\\}$. Then there exist two normal
subgroups $F$ and $L$ of $G$ with the following properties: $F=F_{1}\times
F_{2}$ is nilpotent, where $F_{1}$ and $F_{2}$ are normal subgroups of $G$.
Furthermore, $G/F$ is a Frobenius group with kernel $L/F$, $G/F_{1}$ is a
Frobenius group with kernel $L/F_{1}$, and $G/F_{2}$ is a $2$-Frobenius group.
In particular, $|G/L|$ is an $r$-power, $|L/F_{1}|$ is an $s$-power and
$|F_{1}|$ is an $r$-power. If $r=p$, then $\textbf{O}^{p}(G)\leqslant L$,
$\textbf{O}^{p^{\prime}}(L)\leqslant F_{1}$ and $\textbf{O}^{p}(F_{1})=1$.
Assume that $s=p$. Then $\textbf{O}^{p}(G)=G$,
$\textbf{O}^{p^{\prime}}(G)\leqslant L$, $\textbf{O}^{p}(L)\leqslant F_{1}$
and $\textbf{O}^{p^{\prime}}(F_{1})=1$.

Assume that there exist $a,c\in{\mathrm{Vo}}(G)$ such that $c=p^{m}$ and
$c=p^{k}n$, where $k$, $m$ and $n$ are positive integers and $n\geqslant 2$.
Let $\pi=\\{p,q\\}$. Then the Hall $\pi^{\prime}$-subgroup of $G$ is nilpotent
and normal. By Theorem 2.7, $G$ is a Frobenius group or a nearly $2$-Frobenius
group. Suppose that $G$ is a Frobenius group $KM$ with Frobenius kernel $K$
and Frobenius complement $M$. If $n\mid|M|$, let $r$ be a prime such that
$r\mid n$. Then consider $r$ and $q$. If $s$ is a prime such that $s\mid|K|$,
then by [9, Proposition 3.2], there exists an vanishing element $g$ of $G$
such that ${\mathrm{ord}}(g)$ either divisible by $rs$ or $qs$, a
contradiction. Hence $|M|$ is divisible by two primes $p$ and $q$. Using
Theorem 2.8, we have that $M$ has a unique normal subgroup $N$ such that all
the Sylow subgroups of $N$ are cyclic and
$M/N\in\\{1,\mathrm{V}_{4},\mathrm{A}_{4},\mathrm{S}_{4},\mathrm{A}_{5},\mathrm{S}_{5}\\}$.
Since $G$ is solvable we need not consider the cases when
$M/N\cong\mathrm{A}_{5}$ or $\mathrm{S}_{5}$. Note that $N$ is metacyclic and
supersolvable by [23, p. 290].

Suppose that $M/N\cong 1$. Let $P$ be a Sylow $p$-subgroup of $G$ and $Q$ be a
Sylow $q$-subgroup of $G$. If $p>q$, then by [1, Theorems 6.2.5 and 6.2.2],
$P$ is a normal subgroup of $M$ and hence $KP$ is normal in $G$. Hence
$\textbf{O}^{pp^{\prime}pp^{\prime}}(G)=1$. If $q>p$, then by [1, Theorems
6.2.5 and 6.2.2], $Q$ is a normal subgroup of $M$ and so $K$Q is normal in
$G$. Hence $\textbf{O}^{pp^{\prime}}(G)=1$. Assume that
$M/N\cong\mathrm{V}_{4}$. So $\pi(M)=\\{2,r\\}$, where $r$ is an odd prime.
Let $R$ be a Sylow $r$-subgroup of $G$. Since $N$ is supersolvable, $R$ is
normal in $M$ by [1, Theorems 6.2.5 and 6.2.2]. It follows that $KR$ is normal
in $G$. If $p=2$, then $\textbf{O}^{pp^{\prime}}(G)=1$. If $p=r$, then
$\textbf{O}^{pp^{\prime}pp^{\prime}}(G)=1$. Assume that
$M/N\cong\mathrm{A}_{4}$. Then $\pi(M)=\\{2,3\\}$. If $p=2$, then $N$ is a
$2$-group since the Sylow $3$-subgroup of $M$ is cyclic, and
$\textbf{O}^{pp^{\prime}pp^{\prime}}(G)=1$. If $p=3$, then $N$ is a $3$-group.
This is because if a Sylow $2$-subgroup $T$ has order greater than $4$, then
$T$ is a generalized quaternion and hence has elements of order $2$ and $4$
which are vanishing elements of $G$. Therefore
$\textbf{O}^{pp^{\prime}pp^{\prime}}(G)=1$, as required. Assume that
$M/N\cong\mathrm{S}_{4}$. Then $\pi(M)=\\{2,3\\}$, $p=2$ and $N$ is a
$2$-group. Hence $\textbf{O}^{pp^{\prime}pp^{\prime}}(G)=1$.

Suppose that $G$ is a nearly $2$-Frobenius group. Then there exist two normal
subgroups $F$ and $L$ of $G$ with the following properties: $F=F_{1}\times
F_{2}$ is nilpotent, where $F_{1}$ and $F_{2}$ are normal subgroups of $G$.
Furthermore, $G/F$ is a Frobenius group with kernel $L/F$, $G/F_{1}$ is a
Frobenius group with kernel $L/F_{1}$, and $G/F_{2}$ is a $2$-Frobenius group.
Since $G/F_{2}$ is a $2$-Frobenius group and $G/F$ is a Frobenius group with
kernel $L/F$, it follows that $L/F_{2}$ is a Frobenius group with kernel
$F/F_{2}$. By [9, Remark 1.2], $G/L$ is cyclic and $L/F$ is cyclic with
$|L/F|$ odd. Since $G/F_{1}$ is a Frobenius group with kernel $L/F_{1}$, we
have $\gcd(|G/L|,|L/F_{1}|)=1$. Since $G\setminus
F\subseteq{\mathrm{Van}}(G)$, we have that $|G/L|$ is an $r$-power and $|L/F|$
is $s$-power, where $r$ and $s$ are primes. If $p=r$, then
$\textbf{O}^{p}(G)\leqslant L$, $\textbf{O}^{p^{\prime}}(L)\leqslant F_{1}$.
Hence $\textbf{O}^{pp^{\prime}pp^{\prime}}(G)=1$. If $p=s$, then
$\textbf{O}^{pp^{\prime}}(G)\leqslant L$. Note that the Hall
$p^{\prime}$-subgroup $H$ of $L$ is normal in $L$. Then
$\textbf{O}^{p}(L)\leqslant H$. Hence $\textbf{O}^{p^{\prime}}(H)=1$. Suppose
that $p\nmid a$ for all $a\in{\mathrm{Vo}}(G)$. Then
${\mathrm{Vo}}(G)=\\{a,b\\}$. Either $a$ or $b$ is a prime power by [21,
Theorem B]. Assume that $a$ is an $r$-power. If $b$ is not a prime power, then
the Hall $r^{\prime}$-subgroup is nilpotent and normal in $G$. It follows that
$\textbf{O}^{pp^{\prime}pp^{\prime}}(G)=1$.

Assume that there exists $c\in{\mathrm{Vo}}(G)$ such that $c=p^{k}n$, where
$k$ and $n$ are positive integers and $n\geqslant 2$ and there is no
$a\in{\mathrm{Vo}}(G)$ such that $a=p^{m}$. Then the Hall
$q^{\prime}$-subgroup is normal and nilpotent, and the result follows.

Suppose that $p\nmid a$ for all $a\in{\mathrm{Vo}}(G)$. Then
${\mathrm{Vo}}(G)=\\{a,b\\}$ where $\gcd(a,b)=1$. Either $a$ or $b$ is a prime
power by [21, Theorem B]. We may assume that $b$ is a $q$-power without loss
of generality. Suppose further that $a$ is an $r$-power for some prime $r$.
The Hall $\\{q,r\\}^{\prime}$-subgroup $H$ is nilpotent and normal in $G$.
Then $\textbf{O}^{pp^{\prime}}(G)\leqslant H$ and hence
$\textbf{O}^{pp^{\prime}pp^{\prime}}(G)=1$. If $a$ is not a prime power, then
the Hall $q^{\prime}$-subgroup is nilpotent and normal in $G$. It follows that
$\textbf{O}^{pp^{\prime}pp^{\prime}}(G)=1$. ∎

###### Proof of Theorem C.

Since $p>7$ in the hypothesis of Theorem C, $G$ is solvable using Theorem 3.2.
Now Theorem C follows from Theorem 4.1. ∎

###### Proof of Theorem A.

By Theorem 3.4, $G$ is solvable. Suppose that (a) holds. Then since there are
no $q$-elements that are vanishing elements for all $q\neq p$, the Hall
$p^{\prime}$-subgroup of $G$ is normal and nilpotent by [8, Theorem A]. Hence
$\textbf{O}^{pp^{\prime}}(G)=1$

Suppose that (b) holds. Suppose that $\Gamma(G)$ is connected. Then
${\mathrm{Vo}}(G)=\\{q^{n}\\}$ or ${\mathrm{Vo}}(G)=\\{q^{n}\\}\cup X$, where
$X$ is a finite subset of $\\{c\in\mathbb{N}:pq\mid c\\}$. Then the Hall
$q^{\prime}$-subgroup $H$ is a normal and nilpotent subgroup of $G$ using [8,
Theorem A]. So $\textbf{O}^{pp^{\prime}}(G)\leqslant H$. It follows that
$\textbf{O}^{pp^{\prime}pp^{\prime}}(G)=1$.

We may assume that ${\mathrm{Vo}}(G)=X\cup\\{q^{n}\\}\cup Y$, where $X$ and
$Y$ are finite subsets of $\\{a\in\mathbb{N}:a$ is a $p$-power$\\}$ and
$\\{c\in\mathbb{N}:pq\mid c\\}$, respectively. Let $\pi=\\{p,q\\}$ and suppose
that $P$ is a Sylow $p$-subgroup of $G$. Then the Hall $\pi^{\prime}$-subgroup
$H$ is a normal and nilpotent subgroup of $G$. Then $\overline{G}=G/N$ is a
$\pi$-group, where $N=H\textbf{O}_{p}(G)\textbf{O}_{q}(G)$. Since $Q^{\prime}$
is subnormal, we have that $Q^{\prime}\leqslant\textbf{O}_{q}(G)$ and
$\overline{G}$ is a product of an abelian $q$-group $\overline{Q}=QN/N$ and a
$p$-group $\overline{P}=PN/N$. By Lemma 2.9,
$\overline{Q}\textbf{F}(\overline{G})$ is a normal subgroup of $\overline{G}$.
Then $\overline{G}/\overline{Q}\textbf{F}(\overline{G})$ is a $p$-group and
$\overline{Q}\textbf{F}(\overline{G})/\textbf{O}_{p}(\overline{G})$ is a
$q$-group. Consider $\textbf{O}_{p}(\overline{G})=P_{1}N/N$. Then
$P_{1}N/H\textbf{O}_{q}(G)$ is a $p$-group and $H\textbf{O}_{q}(G)$ is a
$p^{\prime}$-group. Hence $\textbf{O}^{pp^{\prime}pp^{\prime}}(G)=1$.

We may assume that $\Gamma(G)$ is disconnected. Suppose that for all
$a\in{\mathrm{Vo}}(G)\setminus\\{q^{n}\\}$, $a$ is not a $p$-power. Then the
Hall $q^{\prime}$-subgroup $H$ is a normal and nilpotent subgroup of $G$.
Therefore $\textbf{O}^{pp^{\prime}pp^{\prime}}(G)=1$.

Suppose that for all $a\in{\mathrm{Vo}}(G)\setminus\\{q^{n}\\}$, $a$ is a
$p$-power. Then by Theorem D, $G$ is a Frobenius group with a Frobenius
complement of $r$-power order and a kernel of $s$-power order or $G$ is a
nearly $2$-Frobenius group and $\pi(G)=\\{r,s\\}$. Assume that $G$ is a
Frobenius group with a Frobenius complement of $r$-power order and a kernel
$K$ of $s$-power order. If $r=p$, then $\textbf{O}^{pp^{\prime}}(G)=1$. If
$s=p$, then $\textbf{O}^{p}(G)=G$ and $\textbf{O}^{p^{\prime}}(G)=K$. Hence
$\textbf{O}^{p}(K)=1$ and the result follows. Assume that $G$ is a nearly
$2$-Frobenius group and $\pi(G)=\\{r,s\\}$. Then there exist two normal
subgroups $F$ and $L$ of $G$ with the following properties: $F=F_{1}\times
F_{2}$ is nilpotent, where $F_{1}$ and $F_{2}$ are normal subgroups of $G$.
Furthermore, $G/F$ is a Frobenius group with kernel $L/F$, $G/F_{1}$ is a
Frobenius group with kernel $L/F_{1}$, and $G/F_{2}$ is a $2$-Frobenius group.
In particular, $|G/L|$ is an $r$-power, $|L/F_{1}|$ is an $s$-power and
$|F_{1}|$ is an $r$-power. If $r=p$, then $\textbf{O}^{p}(G)\leqslant L$,
$\textbf{O}^{p^{\prime}}(L)\leqslant F_{1}$ and $\textbf{O}^{p}(F_{1})=1$.
Assume that $s=p$. Then $\textbf{O}^{p}(G)=G$,
$\textbf{O}^{p^{\prime}}(G)\leqslant L$, $\textbf{O}^{p}(L)\leqslant F_{1}$
and $\textbf{O}^{p^{\prime}}(F_{1})=1$.

Suppose that there exist $a,b\in{\mathrm{Vo}}(G)\setminus\\{q^{n}\\}$ such
that $a=p^{m}$ and $b=p^{k}t$, where $k$, $m$ and $t$ are positive integers
and $t\geqslant 2$. Let $\pi=\\{p,q\\}$. Using [8, Theorem A], the Hall
$\pi^{\prime}$-subgroup of $G$ is a nilpotent and normal subgroup of $G$. By
Theorem 2.7, $G$ is a Frobenius group or a nearly $2$-Frobenius group. Suppose
that $G$ is a Frobenius group $KM$ with Frobenius kernel $K$ and Frobenius
complement $M$. If $t\mid|M|$, let $r$ be a prime such that $r\mid t$. Since
$M\setminus\\{1\\}\subseteq{\mathrm{Van}}(G)$, $G$ has some $r$-elements which
are vanishing elements. If $r\neq q$, then
$|{\mathrm{Vo}}_{p^{\prime}}(G)|\geqslant 2$, a contradiction. So $r=q$ and
$\Gamma(G)$ is connected, another contradiction from our hypothesis.

Suppose that $G$ is a nearly $2$-Frobenius group. Then there exist two normal
subgroups $F$ and $L$ of $G$ with the following properties: $F=F_{1}\times
F_{2}$ is nilpotent, where $F_{1}$ and $F_{2}$ are normal subgroups of $G$.
Furthermore, $G/F$ is a Frobenius group with kernel $L/F$, $G/F_{1}$ is a
Frobenius group with kernel $L/F_{1}$, and $G/F_{2}$ is a $2$-Frobenius group.
Since $G/F_{2}$ is a $2$-Frobenius group and $G/F$ is a Frobenius group with
kernel $L/F$, it follows that $L/F_{2}$ is a Frobenius group with kernel
$F/F_{2}$. By [9, Remark 1.2], $G/L$ is cyclic and $L/F$ is cyclic with
$|L/F|$ odd. Since $G/F_{1}$ is a Frobenius group with kernel $L/F_{1}$, we
have $\gcd(|G/L|,|L/F_{1}|)=1$. Since $G\setminus
F\subseteq{\mathrm{Van}}(G)$, we have that $|G/L|$ is an $r$-power and $|L/F|$
is $s$-power, where $r$ and $s$ are primes. If $p=r$, then
$\textbf{O}^{p}(G)\leqslant L$, $\textbf{O}^{p^{\prime}}(L)\leqslant F_{1}$.
Hence $\textbf{O}^{pp^{\prime}pp^{\prime}}(G)=1$. If $p=s$, then
$\textbf{O}^{pp^{\prime}}(G)\leqslant L$. Note that the Hall
$p^{\prime}$-subgroup $H$ of $L$ is normal in $L$. Then
$\textbf{O}^{p}(L)\leqslant H$. Hence $\textbf{O}^{p^{\prime}}(H)=1$ and the
result follows. ∎

###### Proof of Theorem B.

We consider the vanishing prime graph $\Gamma(G)$ of $G$. Suppose that
$\Gamma(G)$ is connected. Assume that $\Gamma(G)$ consists of a single vertex.
By Theorem 1.3, $G$ is a $p$-group or $G/\mathbf{Z}(G)$ is a Frobenius group
with a Frobenius complement of $p$-power order and
$\textbf{Z}(G)=\textbf{O}_{p}(G)$. If $G$ is a $p$-group then the result
follows. We may assume that $G/\mathbf{Z}(G)$ is a Frobenius group with a
Frobenius complement of $p$-power order and $\textbf{Z}(G)=\textbf{O}_{p}(G)$.
If $p$ is odd, then Sylow $p$-subgroups are abelian and $P/\textbf{O}_{p}(G)$
is cyclic. If $p=2$, then $P/\textbf{O}_{p}(G)$ is either a generalized
quaternion or cyclic since it is isomorphic to a Frobenius complement. If
$P/\textbf{O}_{p}(G)$ is a generalized quaternion, then $G$ has vanishing
elements of distinct orders divisible by $2$ and $4$, a contradiction. Hence
our result follows.

If $\Gamma(G)$ has more than one vertex, then it follows that the $p$-elements
$G$ are non-vanishing. Hence $P$ is normal in $G$.

We may assume that $\Gamma(G)$ is disconnected. By Theorem 2.7, $G$ is either
a Frobenius or a nearly $2$-Frobenius group. Also note that $p$ is an isolated
vertex.

Suppose that $G$ is a Frobenius group with a kernel $K$ and a complement $H$.
If $p\mid|K|$, then $P$ is normal in $G$ and the result follows. So
$p\mid|H|$. Then either $P$ is a cyclic or a generalized quaternion. If $P$ is
generalized quaternion, then $G$ contains vanishing elements of orders $2$ and
$4$, a contradiction. Then $P$ is cyclic as required.

Suppose that $G$ is a nearly $2$-Frobenius group. Then there exist two normal
subgroups $F$ and $L$ of $G$ with the following properties: $F=F_{1}\times
F_{2}$ is nilpotent, where $F_{1}$ and $F_{2}$ are normal subgroups of $G$.
Furthermore, $G/F$ is a Frobenius group with kernel $L/F$, $G/F_{1}$ is a
Frobenius group with kernel $L/F_{1}$, and $G/F_{2}$ is a $2$-Frobenius group.
Note that $G/L$ and $L/F$ are both cyclic and $F$ is nilpotent. Hence the
result follows. ∎

##  5\. Acknowledgements

The author would like thank the referee for the careful reading of this
article and their comments.

## References

  * [1] H. Bechtell, The theory of groups, New Hampshire, Addison-Wesley, 1971.
  * [2] M. Bianchi, D. Chillag, M. Lewis, and E. Pacifici, Character degree graphs that are complete graphs, _Proc. Amer. Math. Soc._ 135 (2007) 671–676.
  * [3] J. Brough, On vanishing criteria that control finite group structure, _J. Algebra_ 458 (2016) 207–215.
  * [4] R. Brown, Frobenius groups and classical maximal orders, _Mem. Amer. Math. Soc._ 151(717) (2001) viii+110.
  * [5] D. Bubboloni, S. Dolfi and P. Spiga, Finite groups whose irreducible characters vanish only on $p$-elements, _J. Pure Appl. Algebra_ 213 (2009) 370–376.
  * [6] Y. Bugeaud, Z. Cao and M. Mignotte, On simple $K_{4}$-groups, _J. Algebra_ 241 (2001) 658–668.
  * [7] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of Finite Groups, Oxford, Clarendon Press, 1985.
  * [8] S. Dolfi, E. Pacifici, L. Sanus, and P. Spiga, On the orders of zeros of irreducible characters, _J. Algebra_ 321 (2009) 345–352.
  * [9] S. Dolfi, E. Pacifici, L. Sanus, and P. Spiga, On the vanishing prime graph of finite groups, _J. London Math. Soc. (2)_ 82 (2010) 167–183.
  * [10] S. Dolfi, E. Pacifici, L. Sanus, and P. Spiga, On the vanishing prime graph of solvable groups, _J. Group Theory_ 13 (2010) 189–206.
  * [11] L. Dornhoff, Group Representation Theory, Part A: Ordinary Representation Theory, Marcel Dekker, Inc., New York, 1971.
  * [12] S. Franciosi, F. de Giovanni, H. Heineken and M. L. Newell, On the Fitting length of a soluble product of nilpotent groups, _Arch. Math._ 57 (1991) 313–318.
  * [13] E. Gianelli, N. Rizo and A. A. Schaeffer Fry, Groups with few $p^{\prime}$-character degrees, _J. Pure Appl. Algebra_ 224 (2020) 106338.
  * [14] A. Granville, K. Ono, Defect zero $p$-blocks for finite simple groups, _Trans. Amer. Math. Soc._ 348 (1996) 331–347.
  * [15] M. Herzog, On finite simple groups of order divisible by three primes only, _J. Algebra_ 10 (1968) 383–388.
  * [16] G. Higman, Finite groups in which every element has prime power order, _J. London Math. Soc._ 32 (1957) 335–342.
  * [17] I. M. Isaacs, Character Theory of Finite Groups, Rhode Island, Amer. Math. Soc., (2006).
  * [18] I. M. Isaacs, A. Moréto, G. Navarro and P. H. Tiep, Groups with just one character degree divisible by a given prime, _Trans. Amer. Math. Soc._ 361 (2009) 6521–6547.
  * [19] S. Y. Madanha and B. G. Rodrigues, Finite groups with some restriction on the vanishing set, _Comm. Algebra_ 47 (2020) 5474-5481.
  * [20] K. Magaard and H. P. Tong-Viet, Character degree sums in finite non-solvable groups, _J. Group Theory_ 14 (2011) 53–57.
  * [21] G. Malle, G. Navarro and J. B. Olsson, Zeros of characters of finite groups, _J. Group Theory_ 3 (2002) 353–368.
  * [22] G. Qian, Bounding the Fitting height of a finite solvable group by the number of zeros in a character table, _Proc. Amer. Math. Soc._ 130 (2002) 3171–3176.
  * [23] D. J. S. Robinson, A course in the theory of finite groups, Second Edition, New York-Berlin, Springer Verlag, 1995.
  * [24] M. Suzuki, Finite groups with nilpotent centralizers, _Trans. Amer. Math. Soc._ 99 (1961) 425–470.
  * [25] The GAP, (2017), GAP - Groups, Algorithms and Programming, Version 4.8.7. http://www.gap-system.org
  * [26] J. Zhang, Z. Li and C. Shao, Finite groups whose irreducible characters vanish only on elements of prime power order, _International Electronic J. Algebra_ 9 (2011) 114–123.


#  On the universal properties of stochastic processes under optimally tuned
Poisson restart

Sergey Belan<EMAIL_ADDRESS>1Landau Institute for Theoretical Physics,
Russian Academy of Sciences, 1-A Akademika Semenova av., 142432 Chernogolovka,
Russia  2National Research University Higher School of Economics, Faculty of
Mathematics, Usacheva 6, 119048 Moscow, Russia  3National Research University
Higher School of Economics, Faculty of Physics, Myasnitskaya 20, 101000
Moscow, Russia

###### Abstract

Poisson restart assumes that a stochastic process is interrupted and starts
again at random time moments. A number of studies have demonstrated that this
strategy may minimize the expected completion time in some classes of random
search tasks. What is more, it turned out that under optimally tuned restart
rate, any stochastic process, regardless of its nature and statistical
details, satisfies a number of universal relations for the statistical moments
of completion time. In this paper, we describe several new universal
properties of optimally restarted processes. Also we obtain a universal
inequality for the quadratic statistical moments of completion time in the
optimization problem where stochastic process has several possible completion
scenarios.

Introduction. Restarting - the abrupt interruption of a stochastic process
followed by the initiation of a new statistically independent attempt - was
proposed several decades ago in computer science as a method for increasing
the performance of probabilistic algorithms Alt  _et al._ (1991); Luby  _et
al._ (1993). It turned out that if the fluctuations in the completion time of
the randomized algorithm are large enough, then restarts at a properly
selected frequency improves (on average) performance. This effect has a fairly
simple qualitative explanation: if the current implementation of a randomized
algorithm takes too long, then it may be more profitable to abort it and try
again than to wait for the program to terminate.

Later, the potential for increasing the speed of a random process by
restarting it was discovered in the enzymatic kinetics. An elementary act of
an enzymatic reaction begins with the reversible binding of a substrate to an
enzyme. If the intermediate enzyme-substrate is formed, some time is required
for the catalytic step, during which the initial substrate is converted into
products. However, the reversibility of enzyme-substrate binding means that
the intermediate complex can dissociate before the products are formed. In
this case, the next attempt of catalysis can only take place after the enzyme
find another substrate. In other wrds, the dissociation of the intermediate
enzyme-substrate corresponds to restart of the enzymatic reaction. In
theoretical works Reuveni  _et al._ (2014); Rotbart  _et al._ (2015); Reuveni
(2016); Berezhkovskii  _et al._ (2017) it was shown that if the intermediate
complex has several fairly long-lived conformations characterized by
significantly different rates of formation of reaction products, then an
increase in the rate of substrate unbinding may lead to an increase in the
reaction rate.

Depending on the context, it may be interesting to explore different restart
protocols. Say, in computer science, it is often assumed that a randomized
algorithm restarts according to some predetermined schedule Wu and Beek
(2007); Lorenz (2018, 2021); Schulte  _et al._ (2010); Cire  _et al._ (2014);
Amadini  _et al._ (2018); Van Moorsel and Wolter (2006); Wallace and Wallace
(2020); Schroeder and Buro (2001). On the contary, in the case of chemical
transformations, restart events occur at random time moments since decay of
the intermediate complex enzyme-substrate happens due to thermal fluctuations,
see e.g. Reuveni  _et al._ (2014); Rotbart  _et al._ (2015); Reuveni (2016);
Berezhkovskii  _et al._ (2017).

Among many posible restart strategies, stochastic restart at Poisson times is
most extensively studied in the existing literature Evans and Majumdar
(2011a); Evans  _et al._ (2020); Evans and Majumdar (2011b, 2014, 2018);
Blumer  _et al._ (2024); Julián-Salgado  _et al._ (2024); Pal  _et al._
(2022); Pal and Prasad (2019a); Bonomo and Pal (2021); Ray and Reuveni (2020);
Ray  _et al._ (2019). The average value of the random completion time of a
process $T_{r}$, restarted with a rate $r$, is given by a simple formula
Reuveni (2016)

$\langle T_{r}\rangle=\frac{1-\tilde{P}(r)}{r\tilde{P}(r)},$ (1)

where $\tilde{P}(r)=\int_{0}^{\infty}dTP(T)e^{-rT}$ is the Laplace transform
of the completion time probability density $P(T)$ in the absence of restart.

The work of Evans and Majumdar Evans and Majumdar (2011a) showed that there is
an optimal restart rate $r_{\ast}>0$ that minimizes the average completion
time of one-dimensional diffusion-mediated search. Following this, the
possibility of similar optimization was demonstrated for many other random
search problems Evans and Majumdar (2011b); Whitehouse  _et al._ (2013); Evans
and Majumdar (2014); Kusmierz  _et al._ (2014); Kuśmierz and Gudowska-Nowak
(2015); Evans and Majumdar (2018); Ray  _et al._ (2019); Ray and Reuveni
(2020); Singh (2020); Ahmad and Das (2020); Radice (2021); Faisant  _et al._
(2021); Abdoli and Sharma (2021); Santra  _et al._ (2020); Calvert and Evans
(2021); Mercado-Vásquez and Boyer (2021); Bonomo and Pal (2021); Tucci  _et
al._ (2022); Chen and Huang (2022); Ahmad  _et al._ (2022); Pal  _et al._
(2023); Radice (2023). Summarizing the behaviour of the special cases, Reuveni
strictly proved in Ref. Reuveni (2016) that any random process for which there
exists a non-zero optimal restart rate obeys the universal relation

$\frac{\sigma(T_{r_{\ast}})}{\langle T_{r_{\ast}}\rangle}=1,$ (2)

where $\langle T_{r_{\ast}}\rangle$ and $\sigma(T_{r_{\ast}})$ are the average
and standard deviation of the completion time of process $T_{r_{\ast}}$ in the
presence of a restart with optimal rate $r_{\ast}$.

Further, in a recent publication Starkov and Belan (2023) it was demonstrated
that optimally restarted random processes also obey the following universal
inequality

$\frac{\langle T_{r_{\ast}}^{3}\rangle}{\langle T_{r_{\ast}}\rangle^{3}}\geq
6.$ (3)

From the results of the same work it follows that the average completion time
of a random process in the presence of an optimal Poisson restart does not
exceed a quarter of the geometric average completion time of the same process
in the absence of a restart, i.e.

$\langle T_{r_{\ast}}\rangle\geq\frac{1}{4}\langle T^{-1}\rangle^{-1}.$ (4)

Moreover, if the restart-free random completion time $T$ has a unimodal
probability density $P(T)$, then the following inequality constraint is valid
Starkov and Belan (2023)

$\displaystyle\langle T_{r_{\ast}}\rangle\geq\frac{1}{4}M,$ (5)

where $M$ is a mode – the value of the random variable $T$ that occurs most
frequently.

Let us note in pass that the relations (2) and (3) follows from the extremum
conditions $d\langle T_{r}\rangle/dr|_{r=r_{\ast}}=0$ and $d^{2}\langle
T_{r}\rangle/dr^{2}|_{r=r_{\ast}}\geq 0$, respectively. As for the equations
(4) and (5), they are valid not only for the Poisson restart, but for any
restart protocol. Their proofs are based on the special properties of periodic
restart, for more details see Starkov and Belan (2023).

Being insensitive to the process details, the relaitions (2), (3), (4) and (5)
give model-independent predictions regarding the statistical characteristics
of the random processes of different nature under optimally tuned Poisson
restart.

In this work, we derive a number of novel universal relations that govern any
random process at optimal restart conditions. More specifically, we obtain a
lower bound for the optimal rate, an upper bound for the resulting expected
completion time, and a lower bound for the skewness and kurtosis coefficients
of the distribution density of the random completion time. In addition, we
derive a universal inequality in the success probability optimization problem
for a process with two alternative completion scenarios.

Lower bound on the optimal restart rate. If the distribution function $P(T)$
is known, then by minimizing the average completion time $\langle
T_{r}\rangle$, given by the equation (1), with respect to $r$, we can
determine the optimal rate $r_{\ast}$. In a number of cases listed in the work
Rotbart  _et al._ (2015), such a problem allows an exact or approximate
analytical solution.

Is it possible to estimate the optimal rate $r_{\ast}$ without knowing the
detailed statistics of the restarted process? Let us show that the lower bound
on $r_{\ast}$ can be constructed based on information about the first three
statistical moments $\langle T\rangle$, $\langle T^{2}\rangle$ and $\langle
T^{3}\rangle$ (provided they are finite). Indeed, from (1) we find

$\frac{d\langle
T_{r}\rangle}{dr}=\frac{\tilde{P}^{2}(r)-\tilde{P}(r)-r\partial_{r}\tilde{P}(r)}{r^{2}\tilde{P}^{2}(r)}.$
(6)

Estimating the Laplace transform and its derivative using the inequalities
$\tilde{P}(r)\leq 1-r\langle T\rangle+\frac{r^{2}}{2}\langle T^{2}\rangle$
Zubkov (1999) and $\partial_{r}\tilde{P}(r)\geq-\langle T\rangle+r\langle
T^{2}\rangle-\frac{r^{2}}{2}\langle T^{3}\rangle$ Starkov and Belan (2023), we
can write

$\frac{d\langle T_{r}\rangle}{dr}\leq\frac{(\frac{r}{2}\langle
T^{2}\rangle-\langle T\rangle)^{2}+\frac{r}{2}\langle
T^{3}\rangle-\frac{1}{2}\langle T^{2}\rangle}{r^{2}\tilde{P}^{2}(r)}.$ (7)

Next, by solving equation $(\frac{r}{2}\langle T^{2}\rangle-\langle
T\rangle)^{2}+\frac{r}{2}\langle T^{3}\rangle-\frac{1}{2}\langle
T^{2}\rangle\leq 0$ relative to $r$, we find that when $\sigma(T)>\langle
T\rangle$ the average completion time $\langle T_{r}\rangle$ is guaranteed to
decrease on the interval $0<r<\sqrt{\frac{\langle T^{3}\rangle^{2}}{\langle
T^{2}\rangle^{4}}-\frac{4\langle T\rangle\langle T^{3}\rangle}{\langle
T^{2}\rangle^{3}}+\frac{2}{\langle T^{2}\rangle}}-\frac{\langle
T^{3}\rangle}{\langle T^{2}\rangle^{2}}+\frac{2\langle T\rangle}{\langle
T^{2}\rangle}$. From this we can conclude that

$\displaystyle r_{\ast}\geq\sqrt{\frac{\langle T^{3}\rangle^{2}}{\langle
T^{2}\rangle^{4}}-\frac{4\langle T\rangle\langle T^{3}\rangle}{\langle
T^{2}\rangle^{3}}+\frac{2}{\langle T^{2}\rangle}}-\frac{\langle
T^{3}\rangle}{\langle T^{2}\rangle^{2}}+\frac{2\langle T\rangle}{\langle
T^{2}\rangle}.$ (8)

Let us discuss the scope of application of the obtained result. The assumpton
$\sigma(T)>\langle T\rangle$ used in the derivation of Eq. (8) represents a
sufficient (not necessary) condition for the effectiveness of stochastic
restart Rotbart  _et al._ (2015); Reuveni (2016); Pal  _et al._ (2022). This
assumption guarantees that the expression the right side of Eq. (8) is a real
positive quantity. Thus, (8) cannot be used for processes for which
$\sigma(T)<\langle T\rangle$, but nevertheless there is a non-zero restart
rate at which the global minimum of $\langle T_{r}\rangle$ is reached. A
simple example of such a process is given by the following completion time
probability density $P(T)=0.5e^{-rT}+0.5e^{-10rT}$. Next, the formula (8) is
not applicable when at least one of the statistical moments $\langle
T\rangle$, $\langle T\rangle$ and $\langle T^{3}\rangle$ diverges. This is
exactly the case for distributions with fairly heavy tails of completion time
distribution, which is typical for many probabilistic algorithms and random
search processes. Despite the noted shortcomings, inequality(8), to our
knowledge, is the first rigorous result that allows to bound from below the
optimal restart rate based on partial information about the statistics of
original process.

Is it possible, in addition to the lower bound (8), to construct an upper
estimate of the form $r_{\ast}\leq r_{up}$, where $r_{up}$ is expressed in
terms of the first three moments $\langle T\rangle$, $\langle T\rangle$ and
$\langle T^{3}\rangle$? The answer is no since for fixed values of these
moments the optimal rate $r_{\ast}$ can be arbitrarily large, as can be seen
in the following example. Consider, probability density
$P(T)=p\delta(T-t_{1})+(1-p)\delta(T-t_{2})$, where
$t_{2}=\frac{\mu^{2}+\sigma^{2}}{\mu}$,
$p=\frac{\sigma^{2}}{\mu^{2}+\sigma^{2}}$, and $t_{1}\to 0$. It is easy to
check that $\langle T\rangle=\mu$, $\langle T^{2}\rangle=\sigma^{2}+\mu^{2}$
and $\langle T^{3}\rangle=\frac{(\mu^{2}+\sigma^{2})^{2}}{\mu}$. As follows
from (1), if $\sigma>\mu$, then there exists optimal $r_{\ast}\sim
t^{-1}_{1}\to+\infty$ bringing $\langle T_{r}\rangle$ to minimum $\langle
T_{r_{\ast}}\rangle$. Therefore, troika $(\langle T\rangle$,$\langle
T^{2}\rangle$,$\langle T^{3}\rangle)$ does not produce any non trivial upper
bound for $r_{\ast}$.

Bounds on average completion time under optimal restart rate. In addition to
the previously known inequality (4) specifying the lower limit on $\langle
T_{r_{\ast}}\rangle$, it is also possible to determine the upper bound limit.
As was shown in the work Nikitin and Belan (2023), if the condition
$\sigma(T)>\langle T\rangle$ is met, then restart with the rate
$r_{0}=\frac{1}{\langle T\rangle}-\sqrt{\frac{1}{\langle
T\rangle^{2}}-\frac{3\langle T^{2}\rangle}{\langle T\rangle\langle
T^{3}\rangle}+\frac{6\langle T\rangle}{\langle T^{3}\rangle}}$ is guaranteed
to reduce the average process completion time, and $\langle
T_{r_{0}}\rangle\leq\frac{3\langle T^{2}\rangle-2\langle T^{3}\rangle
r_{0}}{6\langle T\rangle}$ . By definition of the optimal rate $\langle
T_{r_{\ast}}\rangle\leq\langle T_{r}\rangle$ for any $r\geq 0$, which means

$\langle T_{r_{\ast}}\rangle\leq\frac{\langle T^{2}\rangle}{2\langle
T\rangle}-\frac{\langle T^{3}\rangle}{3\langle
T\rangle^{2}}+\frac{1}{3}\sqrt{\frac{\langle T^{3}\rangle^{2}}{\langle
T\rangle^{4}}-\frac{3\langle T^{2}\rangle\langle T^{3}\rangle}{\langle
T\rangle^{3}}+\frac{6\langle T^{3}\rangle}{\langle T\rangle}}.$ (9)

This is the upper bound announced just above for the mean completion time in
the presence of an optimally tuned Poisson restart. We emphasize that the
expression in the right side of (9) does not exceed $\langle T\rangle$ if the
sufficient condition for the efficiency of restart $\sigma(T)>\langle
T\rangle$ is satisfied.

Like the formula (8), the relation (9) becomes uninformative in situations
where one or more of the first three statistical moments do not exist. In this
regard, it is of interest to construct an upper bound on $\langle
T_{r_{\ast}}\rangle$, which would be well defined for any process. Let us show
that such a boundary can be expressed in terms of the median completion time
of the unperturbed process $m$, defined by the condition
$\int_{0}^{m}P(T)dT=1/2$.

Since $1-\tilde{P}(r)<1$ and
$\tilde{P}(r)\geq\int_{0}^{r^{-1}}P(T)e^{-rT}dT\geq
e^{-1}\int_{0}^{r^{-1}}P(T)dT$, then taking into account Eq. (1) we obtain
$\langle T_{r}\rangle<\frac{e}{r\int_{0}^{r^{-1}}P(T)dT}$. Substituting
$r=m^{-1}$ and recalling that $\langle T_{r_{\ast}}\rangle\leq\langle
T_{r}\rangle$ for any $r\geq 0$, we finally arrive at the inequality

$\displaystyle\langle T_{r_{\ast}}\rangle\leq 2em.$ (10)

Note that in the work Starkov and Belan (2023) a similar result was obtained
for periodic restart: $\langle T_{\tau_{\ast}}\rangle\leq 2m$, where
$T_{\tau}$ is the random completion time of the process being restarted every
$\tau$ units of time, and $\tau_{\ast}\geq 0$ is a restart period that
minimizes the mean completion time.

The inequality (10) allows us to formulate a new sufficient condition for the
efficiency of Poisson restart. More specifically, if

$\displaystyle\langle T\rangle>2em,$ (11)

then zero rate $r=0$ cannot be optimal, and, therefore, there should exist a
value $r_{\ast}>0$, which reduces the average completion time of the process.
It is easy to see that this criterion is not equivalent to the aforementioned
sufficient condition $\sigma(T)>\langle T\rangle$. Indeed, if
$P(T)=0.6e^{-rT}+0.4e^{-10rT}$, then $\langle T\rangle>2em$, whereas
$\sigma(T)<\langle T\rangle$.

Lower estimates for the skewness and kurtosis coefficients of the random time
distribution density $T_{r_{\ast}}$. Previous studies teach us that optimally
tuned stochastic restart renders very different processes somewhat similar to
each other Reuveni (2016). Namely, as noted above, at the optimal restart rate
$r_{\ast}$ the relative fluctuation of the completion time is equal to unity
for any stochastic process, see Eq. (2). We may recall that unit relative
fluctuation is the property exponentially distributed random variables. This
observation, coupled with the fact that Poisson restart turns the power-law
tails of the completion time probability density into exponential tails Evans
and Majumdar (2011b), may suggest that for any random process under the
optimally tuned restart the statistics of the completion time is exactly
exponential. This conclusion, however, is refuted by counterexamples, see
Reuveni (2016).

Although the complete statistics of the random variable $T_{r_{\ast}}$ is not
universal, its integral characteristics obey some well-defined restrictions.
Let us define a two-dimensional configuration space, where each statistical
distribution law is characterized by two parameters: the skewness and the
kurtosis coefficients, which are defined as standardized moments

$\displaystyle A_{r}=\frac{\langle(T_{r}-\langle
T_{r}\rangle)^{3}\rangle}{\sigma^{3}(T_{r})},$ (12)

and

$K_{r}=\frac{\langle(T_{r}-\langle
T_{r}\rangle)^{4}\rangle}{\sigma^{4}(T_{r})},$ (13)

respectively. The first of these metrics plays the role of the asymmetry
degree of the unimodal probability density relatively to its average value.
The second metric measures the extremality of the deviations of a random
variable from its average value. In descriptive statistics, skewness and
kurtosis are often used to characterize the possible deviations of statistical
data from normality. In our case, it is more interesting to compare with the
case of exponential statistics, for which $A_{exp}=2$ and $K_{exp}=9$. Note
also that for a half-normal distribution one has $A\approx 1$ and
$K_{norm}\approx 3.87$, while in the general case we obtain
$-\infty<A<\infty$, $K\geq A^{2}+1$ Pearson (1916); Sharma and Bhandari
(2015).

Let us demonstrate that relations (2) and (3) impose lower bound on
$A_{r_{\ast}}$ and $K_{r_{\ast}}$. Indeed, from (2) and (12), one finds

$A_{r_{\ast}}=\frac{\langle T_{r_{\ast}}^{3}\rangle-3\langle
T_{r_{\ast}}\rangle\sigma^{2}(T_{r_{\ast}})-\langle
T_{r_{\ast}}\rangle^{3}}{\sigma^{3}(T_{r_{\ast}})}\geq 2.$ (14)

Next, at $r=r_{\ast}$ due to inequality $K_{r}\geq A_{r}^{2}+1$ and Eq. (14)
we obtain

$K_{r_{\ast}}\geq 5.$ (15)

Success probability optimization. Beyond the mean completion time
optimization, stochastic restart can be also used to manipulate the
probability of getting the desired outcome of a random process that has two
alternative completion scenarios – success and failure Belan (2018); Pal and
Prasad (2019b); Singh  _et al._ (2023a); Chechkin and Sokolov (2018); Singh
_et al._ (2023b). For the optimally tuned restart rate $r_{\ast}$, which
minimizes the success probability $p_{r}$, the following universal identity is
valid (see Belan (2018))

$\langle T^{s}_{r_{\ast}}\rangle=\langle T^{f}_{r_{\ast}}\rangle,$ (16)

where $\langle T^{s}_{r_{\ast}}\rangle$ and $\langle T^{f}_{r_{\ast}}\rangle$
denote the average completion time of, respectively, successful and
unsuccessful process realizations. Below we show that in addition to Eq. (16),
the following inequality for the quadratic statistical moments of completion
times is also valid at the optimal restart rate

$\langle{T^{s}_{r_{\ast}}}^{2}\rangle\leq\langle{T^{f}_{r_{\ast}}}^{2}\rangle.$
(17)

The completion time probability density of a random process with two competing
outcomes can be represented as $P(T)=P^{s}(T)+P^{f}(T)$, where $P^{s}(T)$ and
$P^{f}(T)$ correspond the contributions of successful and unsuccessful trials,
respectively. The restart-free probability of success is then expressed as
$p=\int_{0}^{\infty}P^{s}(T)dT$, and the mean time for successful runs is
given by $\langle T^{s}\rangle=p^{-1}\int_{0}^{\infty}P^{s}(T)TdT$. In the
presence of stochastic restart with rate $r$, the probability of success
becomes

$p_{r}=\frac{\tilde{P}^{s}(r)}{\tilde{P}(r)}.$ (18)

Here $\tilde{P}^{s}(r)$ and $\tilde{P}(r)$ denote the Laplace-transforms of
$P^{s}(T)$ and $P(T)$ evaluated at $r$. The relation (16) follows from Eq.(18)
and extremality condition $dp_{r}/dr|_{r=r_{\ast}}=0$.

Now assume that a process which is already restarted at optimal rate
$r_{\ast}$ becomes additionally subject to differentially small retart rate
$\delta r$. Due to the additivity of Poisson flows and the condition
$dp_{r_{\ast}}/dr|_{r_{\ast}}=0$, we can write

$p_{r_{\ast}+\delta r}=p_{r_{\ast}}+\frac{\delta
r^{2}}{2}\frac{d^{2}p_{r}}{dr^{2}}|_{r_{\ast}}+o(\delta r^{2}).$ (19)

At the same time, from (18) we find

$\displaystyle p_{r_{\ast}+\delta r}=\frac{\tilde{P}^{s}_{r_{\ast}}(\delta
r)}{\tilde{P}_{r_{\ast}}(\delta r)}=\frac{\int_{0}^{\infty}dTe^{-\delta
rT}P_{r_{\ast}}^{s}(T)}{\int_{0}^{\infty}dTe^{-\delta rT}P_{r_{\ast}}(T)}=$
(20) $\displaystyle=\frac{1-\delta r\langle
T^{s}_{r_{\ast}}\rangle+\frac{\delta r^{2}}{2}\langle
T^{s2}_{r_{\ast}})+o(\delta r^{2})}{1-\delta r\langle
T_{r_{\ast}}\rangle+\frac{\delta r^{2}}{2}\langle
T^{2}_{r_{\ast}}\rangle+o(\delta r^{2})}p=$ (21)
$\displaystyle=p_{r_{\ast}}+\frac{\delta
r^{2}}{2}p_{r_{\ast}}(1-p_{r_{\ast}})(\langle{T_{r_{\ast}}^{s}}^{2}\rangle-\langle{T_{r_{\ast}}^{f}}^{2}\rangle)+o(\delta
r^{2}),$ (22)

where we used an obvious identity $\langle
T^{2}_{r}\rangle=p\langle{T_{r}^{s}}^{2}\rangle+(1-p)\langle{T_{r}^{f}}^{2}\rangle$.
Comparing (19) and (22), we find
$d^{2}p_{r_{\ast}}/dr^{2}|_{r_{\ast}}=p_{r_{\ast}}(1-p_{r_{\ast}})(\langle{T_{r_{\ast}}^{s}}^{2}\rangle-\langle{T_{r_{\ast}}^{f}}^{2}\rangle)/2$.
The point of maximum obyes the condition
$d^{2}p_{r_{\ast}}/dr^{2}|_{r=r_{\ast}}\leq 0$, which immediately yields Eq.
(17). Thus, under conditions of optimal stochastic restart, the completion
time of successful trials is characterized by smaller fluctuations compared to
unsuccessful attempts.

Conclusion. Our analysis contributes to the understanding of the universal
features inherent to random processes subject to optimal Poisson restart. The
main results of the work are given by the relations (8), (9), (10), (14), (15)
and (17). As a useful consequence of these results, we have obtained a new
criterion for the efficiency of Poisson restart, see Eq. (11). Future research
should answer the question of whether the limits set by our inequalities are
achieved. In addition, a practically important question is how the presented
results will change when taking into account the non-zero time required to
perform the restart.

###### Acknowledgements.

The work was supported by the Russian Science Foundation, project no.
22-72-10052.

## References

  * Alt  _et al._ (1991) H. Alt, L. Guibas, K. Mehlhorn, R. Karp, and A. Wigderson, Technical Report TR-91-057 (1991).
  * Luby  _et al._ (1993) M. Luby, A. Sinclair, and D. Zuckerman, Information Processing Letters 47, 173 (1993).
  * Reuveni  _et al._ (2014) S. Reuveni, M. Urbakh, and J. Klafter, Proceedings of the National Academy of Sciences 111, 4391 (2014).
  * Rotbart  _et al._ (2015) T. Rotbart, S. Reuveni, and M. Urbakh, Physical Review E 92, 060101 (2015).
  * Reuveni (2016) S. Reuveni, Physical Review Letters 116, 170601 (2016).
  * Berezhkovskii  _et al._ (2017) A. M. Berezhkovskii, L. Dagdug, and S. M. Bezrukov, The Journal of Chemical Physics 147 (2017).
  * Wu and Beek (2007) H. Wu and P. Beek, In International Conference on Principles and Practice of Constraint Programming , 681 (2007).
  * Lorenz (2018) J. Lorenz, In International Conference on Current Trends in Theory and Practice of Informatics , 493 (2018).
  * Lorenz (2021) J. Lorenz, Theory of Computing Systems 65, 1143 (2021).
  * Schulte  _et al._ (2010) C. Schulte, G. Tack, and M. Z. Lagerkvist, Schulte, Christian and Tack, Guido and Lagerkvist, Mikael 1 (2010).
  * Cire  _et al._ (2014) A. Cire, S. Kadioglu, and M. Sellmann, in  _Proceedings of the AAAI Conference on Artificial Intelligence_ , Vol. 28 (2014).
  * Amadini  _et al._ (2018) R. Amadini, M. Gabbrielli, and J. Mauro, Theory and Practice of Logic Programming 18, 81 (2018).
  * Van Moorsel and Wolter (2006) A. P. Van Moorsel and K. Wolter, IEEE Transactions on Software Engineering 32, 547 (2006).
  * Wallace and Wallace (2020) M. Wallace and M. Wallace, Building Decision Support Systems: using MiniZinc , 165 (2020).
  * Schroeder and Buro (2001) M. Schroeder and L. Buro, in  _Proceedings of the Workshop on Infrastructure for Agents, MAS, and Scalable MAS at the Conference Autonomous Agents_ (2001).
  * Evans and Majumdar (2011a) M. Evans and S. Majumdar, Physical Review Letters 106, 160601 (2011a).
  * Evans  _et al._ (2020) M. R. Evans, S. N. Majumdar, and G. Schehr, Journal of Physics A: Mathematical and Theoretical 53, 193001 (2020).
  * Evans and Majumdar (2011b) M. R. Evans and S. N. Majumdar, Journal of Physics A: Mathematical and Theoretical 44, 435001 (2011b).
  * Evans and Majumdar (2014) M. R. Evans and S. N. Majumdar, Journal of Physics A: Mathematical and Theoretical 47, 285001 (2014).
  * Evans and Majumdar (2018) M. R. Evans and S. N. Majumdar, Journal of Physics A: Mathematical and Theoretical 51, 475003 (2018).
  * Blumer  _et al._ (2024) O. Blumer, S. Reuveni, and B. Hirshberg, Nature Communications 15, 240 (2024).
  * Julián-Salgado  _et al._ (2024) P. Julián-Salgado, L. Dagdug, and D. Boyer, Physical Review E 109, 024134 (2024).
  * Pal  _et al._ (2022) A. Pal, S. Kostinski, and S. Reuveni, Journal of Physics A: Mathematical and Theoretical 55, 021001 (2022).
  * Pal and Prasad (2019a) A. Pal and V. Prasad, Physical Review E 99, 032123 (2019a).
  * Bonomo and Pal (2021) O. L. Bonomo and A. Pal, Physical Review E 103, 052129 (2021).
  * Ray and Reuveni (2020) S. Ray and S. Reuveni, The Journal of chemical physics 152 (2020).
  * Ray  _et al._ (2019) S. Ray, D. Mondal, and S. Reuveni, Journal of Physics A: Mathematical and Theoretical 52, 255002 (2019).
  * Whitehouse  _et al._ (2013) J. Whitehouse, M. R. Evans, and S. N. Majumdar, Physical Review E 87, 022118 (2013).
  * Kusmierz  _et al._ (2014) L. Kusmierz, S. N. Majumdar, S. Sabhapandit, and G. Schehr, Physical review letters 113, 220602 (2014).
  * Kuśmierz and Gudowska-Nowak (2015) Ł. Kuśmierz and E. Gudowska-Nowak, Physical Review E 92, 052127 (2015).
  * Singh (2020) P. Singh, Journal of Physics A: Mathematical and Theoretical 53, 405005 (2020).
  * Ahmad and Das (2020) S. Ahmad and D. Das, Physical Review E 102, 032145 (2020).
  * Radice (2021) M. Radice, Physical Review E 104, 044126 (2021).
  * Faisant  _et al._ (2021) F. Faisant, B. Besga, A. Petrosyan, S. Ciliberto, and S. N. Majumdar, Journal of Statistical Mechanics: Theory and Experiment 2021, 113203 (2021).
  * Abdoli and Sharma (2021) I. Abdoli and A. Sharma, Soft Matter 17, 1307 (2021).
  * Santra  _et al._ (2020) I. Santra, U. Basu, and S. Sabhapandit, Journal of Statistical Mechanics: Theory and Experiment 2020, 113206 (2020).
  * Calvert and Evans (2021) G. R. Calvert and M. R. Evans, The European Physical Journal B 94, 1 (2021).
  * Mercado-Vásquez and Boyer (2021) G. Mercado-Vásquez and D. Boyer, Journal of Physics A: Mathematical and Theoretical 54, 444002 (2021).
  * Tucci  _et al._ (2022) G. Tucci, A. Gambassi, S. N. Majumdar, and G. Schehr, Physical Review E 106, 044127 (2022).
  * Chen and Huang (2022) H. Chen and F. Huang, Physical Review E 105, 034109 (2022).
  * Ahmad  _et al._ (2022) S. Ahmad, K. Rijal, and D. Das, Physical Review E 105, 044134 (2022).
  * Pal  _et al._ (2023) A. Pal, V. Stojkoski, and T. Sandev, arXiv preprint arXiv:2310.12057 (2023).
  * Radice (2023) M. Radice, Physical Review E 107, 024136 (2023).
  * Starkov and Belan (2023) D. Starkov and S. Belan, Physical Review E 107, L062101 (2023).
  * Zubkov (1999) A. Zubkov, Theory of Probability & Its Applications 43, 676 (1999).
  * Nikitin and Belan (2023) I. Nikitin and S. Belan, arXiv preprint arXiv:2309.05877 (2023).
  * Pearson (1916) K. Pearson, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character 216, 429 (1916).
  * Sharma and Bhandari (2015) R. Sharma and R. Bhandari, (2015).
  * Belan (2018) S. Belan, Physical Review Letters 120, 080601 (2018).
  * Pal and Prasad (2019b) A. Pal and V. Prasad, Physical Review E 99, 032123 (2019b).
  * Singh  _et al._ (2023a) R. Singh, T. Sandev, and S. Singh, arXiv preprint arXiv:2305.01601 (2023a).
  * Chechkin and Sokolov (2018) A. Chechkin and I. Sokolov, Physical review letters 121, 050601 (2018).
  * Singh  _et al._ (2023b) R. Singh, T. Sandev, and S. Singh, Physical Review E 108, L052106 (2023b).