# On almost $p$-rational characters in principal blocks

Attila Maróti, J. Miquel Martínez, A. A. Schaeffer Fry and Carolina Vallejo

###### Abstract.

Let $p$ be a prime. In this paper we provide a lower bound for the number of
almost $p$-rational characters of degree coprime to $p$ in the principal
$p$-block of a finite group of order divisible by $p$. We further describe the
$p$-local structure of the groups for which the above-mentioned bound is
sharp.

###### Key words and phrases:

Galois action on characters, principal blocks

###### 1991 Mathematics Subject Classification:

$20$C$15$ ($20$C$20$)

The first-named author was supported by the National Research, Development and
Innovation Office (NKFIH) Grant No. K138596, No. K132951 and Grant No.
K138828. The second-named author is funded by the national project PRIN 2022-
2022PSTWLB - Group Theory and Applications - CUP B53D23009410006. The third-
named author gratefully acknowledges support from the National Science
Foundation, Award No. DMS-2100912, and her former institution, Metropolitan
State University of Denver, which holds the award and allows her to serve as
PI. The fourth-named author, as part of the GNSAGA, is grateful for the
support of the Istituto Nazionale di Alta Matematica (INdAM). Part of this
work was carried out while the second-named author visited the University of
Florence during the fall of 2023, funded by a Borsa Ferran Sunyer i Balaguer
2023 from l’Institut d’Estudis Catalans. He thanks l’Institut for the support
and the Dipartamento di Matematica e Informatica ‘Ulisse Dini’ for the warm
hospitality.

##  1\. Introduction

Let $G$ be a finite group and let $p$ be a prime dividing $|G|$. Let $\xi$ be
a primitive $|G|$th root of unity. Write $|G|=p^{a}n$ with $(p,n)=1$ and $a$
is a positive integer. Let $\sigma\in{\rm
Gal}({\mathbb{Q}}(\xi)/{\mathbb{Q}})$ be the Galois automorphism that sends
$p$-power roots of unity to their $1+p$ power and fixes roots of unity of
order coprime to $p$. Although the definition of $\sigma$ depends on the prime
$p$, it will be clear from context throughout the article. Given a positive
integer $k$, we will denote by ${\mathbb{Q}}_{k}$ the $k$th cyclotomic
extension of ${\mathbb{Q}}$. Following [HMM22], we say that a character $\chi$
is almost $p$-rational if all the values of $\chi$ lie in some cyclotomic
extension ${\mathbb{Q}}_{pm}$ where $(p,m)=1$. If $p$ is odd, then ${\rm
Gal}({\mathbb{Q}}_{|G|}/{\mathbb{Q}}_{pn})=\langle\sigma\rangle$. In
particular, a character $\chi$ of $G$ is almost $p$-rational if, and only if,
$\chi$ is $\sigma$-invariant. If $p=2$, then
${\mathbb{Q}}_{2n}={\mathbb{Q}}_{n}$, so almost $2$-rational characters are
$2$-rational. As ${\rm Gal}({\mathbb{Q}}_{|G|}/{\mathbb{Q}}_{n})$ is not
generally cyclic, we cannot characterize $2$-rational characters as those
invariant under the action of $\langle\sigma\rangle$ (for instance, dihedral
groups of order $2^{k}>8$ have irreducible characters that are
$\sigma$-invariant but not 2-rational). However, when dealing with irreducible
characters $\chi$ of odd degree, it was first noticed in [VR23] that $\chi$ is
$2$-rational if, and only if, $\chi$ is $\sigma$-invariant (or equivalently
$\langle\sigma\rangle$-fixed).

Let $G$ be a finite group and let $p$ be a prime. For a $p$-block $B$ of $G$,
let $k(B)$ denote the number of complex irreducible characters in $B$. In
[HK00], Héthelyi and Külshammer conjecture that if $k(B)>1$ then

$k(B)\geq 2\sqrt{p-1}.$

This conjecture remains open even for solvable groups. Recently, Hung and the
third-named author of this article have shown that the Héthelyi–Külshammer
conjecture holds for principal blocks in [HS23]. Our aim in this article is to
show that a much stronger bound is true for principal blocks. More precisely,
we show that $2\sqrt{p-1}$ is also a lower bound for the number of almost
$p$-rational characters of degree coprime to $p$ lying in the principal
$p$-block of groups of order divisible by $p$. From the discussion above, this
means we are interested in studying

$k_{0,\sigma}(B_{0}(G)),$

the size of the set
$\displaystyle\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(G))$ of irreducible
characters of degree coprime to $p$ in principal $p$-blocks of $G$ fixed by
the action of $\langle\sigma\rangle$. We care to remark that the numbers
$k_{0,\sigma}(B_{0}(G))$ have been recently shown to influence the numbers of
generators of the Sylow $p$-subgroups in [NT19, RSV20, VR23]. We denote by
$\Phi(G)$ the Frattini subgroup of a group $G$ and by $\lceil
x\rceil\in\mathbb{N}$ the ceiling of a real number $x$. Our main result is the
following.

######  Theorem A.

Let $G$ be a finite group, $p$ a prime and $P\in\operatorname{Syl}_{p}(G)$. If
$P>1$, then

$\displaystyle k_{0,\sigma}(B_{0}(G))\geq\lceil 2\sqrt{p-1}\rceil\,$

with equality if, and only if, the following conditions hold:

  1. (i)

$\lceil 2\sqrt{p-1}\rceil=e+\frac{p-1}{e}$ for some divisor $e$ of $p-1$;

  2. (ii)

$P$ is cyclic; and

  3. (iii)

the local quotient group ${\bf N}_{G}(P)/\Phi(P){\bf O}_{p^{\prime}}({\bf
N}_{G}(P))$ is isomorphic to one of the following Frobenius groups
$\displaystyle\mathsf{C}_{p}\rtimes\mathsf{C}_{e}$ or
$\displaystyle\mathsf{C}_{p}\rtimes\mathsf{C}_{\frac{p-1}{e}}$.

The statement of Theorem A can be proven to be a consequence of the
Alperin–McKay–Navarro conjecture [Nav04, Conjecture B]. In this sense, Theorem
A provides new evidence of the validity of this conjecture, which as of the
writing of this article is quite far from being proved. (For example, it has
not yet been reduced to a problem on simple groups.) We give further details
in Section 7.2, where we also speculate about the possibility of extending the
inequality in Theorem A to arbitrary blocks.

Given any finite group $G$ of order divisible by a prime $p$, it was proven in
[Mar16], by the first-named author of this article, that the number $k(G)$ of
conjugacy classes of $G$ (or equivalently the number of irreducible complex
characters of $G$) is bounded by below by $2\sqrt{p-1}$. Moreover, the results
of [Mar16] show that the bound $k(G)=2\sqrt{p-1}$ is attained only if
$a=\sqrt{p-1}\in\mathbb{N}$ and $G$ is isomorphic to the Frobenius group ${\sf
C}_{p}\rtimes{\sf C}_{a}$. Our proof of the equality in Theorem A ultimately
depends on the analysis of the equality $k(G)=\lceil 2\sqrt{p-1}\rceil$ in
groups $G$ with a nontrivial normal elementary abelian Sylow $p$-subgroup. We
carry out this analysis, improving upon [Mar16, Theorem 1.1], in Section 2.
Additionally, our proof of Theorem A depends on the following statements
regarding mostly finite simple groups.

######  Theorem B.

Let $S$ be a finite nonabelian simple group of order divisible by a prime
$p\geq 5$. Let $A=\operatorname{Aut}(S)$.

  1. (i)

There is some $X$-invariant
$1_{S}\neq\theta\in\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(S))$, where
$X/S\in\operatorname{Syl}_{p}(A/S)$.

  2. (ii)

If $S$ has non-cyclic Sylow $p$-subgroups and $S\leq T\leq A$ with
$p\nmid|T:S|$ then

    1. (a)

$k_{0,\sigma}(B_{0}(T))>\lceil 2\sqrt{p-1}\rceil$,

    2. (b)

the number of $T$-orbits on $\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(S))$
is at least $2(p-1)^{1/4}$.

  3. (iii)

Suppose that $S$ has cyclic Sylow $p$-subgroups. Write $N=S^{t}$ with $t\geq
2$ and suppose that $N\triangleleft\,H$ with $p\nmid|H:N|$. Then

$k_{0,\sigma}(B_{0}(H))=k_{0,\sigma}(B_{0}({\bf N}_{H}(P)))>\lceil
2\sqrt{p-1}\rceil$

where $P\in\operatorname{Syl}_{p}({H})$.

We note that Theorem B(i) implies that there is a $\sigma$-invariant extension
of $\theta$ to $X$, using [Nav18, Cors. 6.2 and 6.4]. With this in mind, we
prove a stronger statement in the case that $S$ is a simple groups of Lie type
in non-defining characteristic in Lemma 4.4 below, which might be useful for
related future applications.

Notice also that Theorem B(iii) states that the principal $p$-block of a group
$H$ as described in its statement satisfies [IN02, Conjecture D]. In [HMM22,
Theorem 4.4], the authors prove a weaker version of the statement of Theorem
B(iii) by using the Classification of Finite Simple Groups (CFSG, for short)
when proving that $2\sqrt{p-1}$ is a lower bound for the number of almost
$p$-rational irreducible characters of degree coprime to $p$ in a group of
order divisible by $p$. Remarkably, we prove Theorem B(iii) without invoking
the CFSG in Section 5.

This paper is organized as follows. In Section 2 we show that if $G$ is a
finite group of order divisible by $p$ and $k(G)=\lceil 2\sqrt{p-1}\rceil$
then $G$ has cyclic Sylow $p$-subgroups. Section 3 contains known results on
principal blocks. We prove Theorem B(i) and (ii) in Section 4 and Theorem
B(iii) in Section 5. We complete the proof of Theorem A in Section 6. In
Section 7 we discuss some related problems.

##  2\. An improved bound on the number of conjugacy classes

The aim of this section is to characterize the finite groups $G$ with an
elementary abelian nontrivial Sylow $p$-subgroup such that $k(G)=\lceil
2\sqrt{p-1}\rceil$. By [Mar16], the inequality $k(G)\geq\lceil
2\sqrt{p-1}\rceil$ always holds. We will show that $k(G)=\lceil
2\sqrt{p-1}\rceil$ implies that $|G|_{p}=p$, where $n_{p}$ is the $p$-part of
the number $n$. In particular, $k(G)=a+\frac{p-1}{a}$ for some divisor $a$ of
$p-1$, and $G$ is isomorphic to one of the Frobenius groups ${\sf
C}_{p}\rtimes{\sf C}_{a}$ or $\displaystyle{\sf C}_{p}\rtimes{\sf
C}_{\frac{p-1}{a}}$.

###  2.1. Statement of results

Let $G$ be a finite group and $p$ a prime. Let $k(G)$ be the number of
conjugacy classes of $G$. For a normal subgroup $V$ in $G$, let $n(G,V)$
denote the number of orbits of $G$ on $V$ (acting by conjugation).

The purpose of this section is to prove the following.

######  Theorem 2.1.

If $G$ is a finite group having an elementary abelian minimal normal subgroup
$V$ of $p$-rank at least $2$ and $|G/V|$ is not divisible by the prime $p$,
then $k(G)\geq 2\sqrt{p-1}+1$.

We make a few remarks on Theorem 2.1.

In order to prove Theorem 2.1, we may assume that $G$ is not nilpotent. For if
$G$ is nilpotent, then $k(G)\geq|V|\geq p^{2}\geq 2\sqrt{p-1}+1$. In the paper
[VLS07] and its prequels, all non-nilpotent finite groups are classified with
at most $14$ conjugacy classes. By going through these lists of groups, we see
that no group $G$ is a counterexample to Theorem 2.1 with $k(G)\leq 14$. This
means that in Theorem 2.1 we can assume that $2\sqrt{p-1}+1\geq 15$. In other
words, $p\geq 53$.

Héthelyi and Külshammer [HK03] proved that if $X$ is a finite solvable group
whose order is divisible by the square of a prime $p$ then
$k(X)\geq(49p+1)/60$. Thus if $G$ is solvable (and $p\geq 53$), then
$k(G)\geq(49p+1)/60\geq 2\sqrt{p-1}+1$. We may assume in Theorem 2.1 that $G$
is not solvable.

Since $k(G)\geq k(G/V)+n(G,V)-1$ by the Clifford-Gallagher formula, it is
sufficient to show that $k(G/V)+n(G,V)\geq 2\sqrt{p-1}+2$. Since $k(G/V)\geq
k(G/{\bf C}_{G}(V))$ and $n(G,V)=n(G/{\bf C}_{G}(V),V)$, we may assume that
$G/V$ acts faithfully on $V$, that is, $V$ is a faithful and irreducible
$G/V$-module.

We find that Theorem 2.1 would be a consequence of the following statement
(where the new $G$ is $G/V$ in Theorem 2.1).

######  Theorem 2.2.

Let $V$ be an irreducible and faithful $FG$-module for some finite non-
solvable group $G$ and finite field $F$ of characteristic $p$ at least $53$.
If $p$ does not divide $|G|$, then $k(G)+n(G,V)-1\geq 2\sqrt{p-1}+1$.

###  2.2. Preliminaries

The size of the field $F$ will be denoted by $q$, the dimension of $V$ over
$F$ by $n$, and the center of ${\mathrm{GL}}(n,q)$ by $Z$.

We will use the following trivial observation throughout the paper.

######  Lemma 2.3.

With the notation and assumptions above, $|V|/|G|\leq n(G,V)$.

Suppose that $G$ transitively permutes a set $\\{V_{1},\ldots,V_{t}\\}$ of
subspaces of $V$ with $t$ an integer with $1\leq t\leq n$ as large as possible
with the property that $V=V_{1}\oplus\cdots\oplus V_{t}$. Let $B$ be the
kernel of this action of $G$ on the set of subspaces. Note that $G/B$ is a
transitive permutation group of degree $t$. The subgroup $B$ is isomorphic to
a subdirect product of $t$ copies of a finite group $T$. In other words $B$ is
isomorphic to a subgroup of $T_{1}\times\cdots\times T_{t}$ where for each $i$
with $1\leq i\leq t$ the vector space $V_{i}$ is a faithful $T_{i}$-module and
$T_{i}\cong T$. Let $H_{1}$ be the stabilizer of $V_{1}$ in $G$. Let $k$ be
the number of orbits of $H_{1}$ on $V_{1}$.

The following is [Fou69, Lemma 2.6].

######  Lemma 2.4.

With the above notation and assumptions,

$\max\\{t+1,k\\}\leq\binom{t+k-1}{k-1}\leq n(G,V).$

We will use the following consequence of a result of Seager [Sea87, Theorem
1].

######  Proposition 2.5.

Let $W$ be a faithful primitive $FH$-module for a finite solvable group $H$
not contained in $\mathrm{\Gamma L}(1,p^{m})$ where $F$ is a field of prime
order $p\geq 53$ and $|W|=p^{m}$. Then $p^{m/2}/12m<n(H,W)$.

The following is [PP98, Proposition 4].

######  Proposition 2.6.

Let $X$ be any subgroup of the symmetric group $S_{m}$ whose order is coprime
to a prime $p$. If $m>1$ then $|X|<p^{m-1}$.

The proof of the next lemma is identical to the proof of [Mar16, Lemma 3.5].

######  Lemma 2.7.

If $G$ has an abelian subgroup of index at most ${|V|}^{1/2}/(2\sqrt{p-1}+1)$
and $n(G,V)\leq 2\sqrt{p-1}+1$, then $k(G)\geq 2\sqrt{p-1}+1$.

###  2.3. The class $\mathcal{C}_{q}$

Our first aim in proving Theorem 2.2 is to describe (as much as possible) the
possibilities for $G$ and $V$ with the condition that $n(G,V)<2\sqrt{q-1}+1$
where $q$ is the size of the underlying field $F$. For this we need to
introduce a class of pairs $(G,V)$ which we denote by $\mathcal{C}_{q}$.

In this paragraph we define a class of pairs $(G,V)$ where $V$ is an
$FG$-module. Let $W$ be a not necessarily faithful but coprime $QH$-module for
some finite field extension $Q$ of $F$ and some finite group $H$. We write
$\mathrm{Stab}_{Q_{1}}^{Q}(H,W)$ for the class of pairs $(H_{1},W_{1})$ with
the property that $W_{1}$ is a $Q_{1}H_{1}$-module with $F\leq Q_{1}\leq Q$
where $W_{1}$ is just $W$ viewed as a $Q_{1}$-vector space and $H_{1}$ is some
group with the following property. If
$\varphi:H_{1}\rightarrow{\mathrm{GL}}(W_{1})$ and
$\psi:H\rightarrow{\mathrm{GL}}(W)$ denote the natural, not necessarily
injective homomorphisms, then $\varphi(H_{1})\cap{\mathrm{GL}}(W)=\psi(H)$. We
write $\mathrm{Ind}(H,W)$ for the class of pairs $(H_{1},W_{1})$ with the
property that $W_{1}=\mathrm{Ind}_{H}^{H_{1}}(W)$ for some group $H_{1}$ with
$H\leq H_{1}$. Finally, let $\mathcal{C}_{q}$ be the class of all pairs
$(G,V)$ with the property that $V$ is a finite, faithful, coprime and
irreducible $FG$-module so that $(G,V)$ can be obtained by repeated
applications of $\mathrm{Stab}_{Q_{1}}^{Q_{2}}$ and $\mathrm{Ind}$ starting
with $(H,W)$ where $W$ is a $1$-dimensional $QH$-module with $Q$ a field
extension of $F$.

The main results of this section are Lemmas 2.8 and 2.9.

######  Lemma 2.8.

Let $(G,V)\in\mathcal{C}_{q}$ and $n(G,V)<2\sqrt{q-1}+1$. If $q\geq 53$, then
$|G|<{|V|}^{3/2}$.

Lemma 2.8 is a slight variation of Lemma 4.1 of [Mar16]. The proof of Lemma
2.8 may be obtained from the proof of Lemma 4.1 of [Mar16] by minor changes in
places (and using Lemma 2.4 and Proposition 2.6).

After minor changes to the proof of Lemma 4.2 of [Mar16], the condition $p\geq
59$ may be relaxed to $p\geq 53$ in the statement and the two occurrences of
$2\sqrt{p-1}$ may be changed to $2\sqrt{p-1}+1$. We obtain the following.

######  Lemma 2.9.

Let $(G,V)\in\mathcal{C}_{q}$ and $n(G,V)<2\sqrt{q-1}+1$. If $p\geq 53$, then
at least one of the following holds.

  1. (i)

$G$ has an abelian subgroup of index at most ${|V|}^{1/2}/(2\sqrt{p-1}+1)$.

  2. (ii)

$|F|=p$, the module $V$ is induced from a $1$-dimensional module, and $G$ has
a factor group isomorphic to $\mathfrak{A}_{n}$ or $\mathfrak{S}_{n}$ where
$n=\dim_{F}(V)$. In this case we either have $n=1$, or $15\leq n\leq 180$ and
$p<8192$.

###  2.4. Some absolutely irreducible representations

The following is Proposition 5.1 of [Mar16] with two occurrences of $59$
changed to $53$ and one occurrence of $2\sqrt{q-1}$ changed to
$2\sqrt{q-1}+1$.

######  Proposition 2.10.

Suppose that $p$ is a prime at least $53$. Let $H$ be a finite subgroup of
${\mathrm{GL}}(n,q)$ with generalized Fitting subgroup a quasisimple group
where $q$ is a power of $p$. Put $G=Z\circ H$ where $Z$ is the multiplicative
group of $F$. Furthermore suppose that $V$ is an absolutely irreducible
$FT$-module for every non-central normal subgroup $T$ of $G$. Suppose also
that $|G|$ is not divisible by $p$. Then $n(G,V)\geq 2\sqrt{q-1}+1$ unless
possibly if $n=2$, $q$ is in the range $53\leq q\leq 14389$, it is congruent
to $\pm 1$ modulo $10$, and $G=Z\circ 2.\mathfrak{A}_{5}$.

Suppose that the group $G$ has a unique normal subgroup $R$ which is minimal
subject to being non-central. Suppose that $R$ is an $r$-group of symplectic
type for some prime $r$ (this is an $r$-group all of whose characteristic
abelian subgroups are cyclic). Suppose that $V$ is an absolutely irreducible
$FR$-module. Let $|R/Z(R)|=r^{2a}$ for some positive integer $a$. Then the
dimension of the module is $n=r^{a}$. Suppose that $Z\leq G$. The group
$G/(RZ)$ can be considered as a subgroup of the symplectic group
${\mathrm{Sp}}_{2a}{(r)}$. As always, we assume that $q\geq p\geq 53$.

The following is Proposition 5.2 of [Mar16] with the obvious changes ($59$
became $53$ and $2\sqrt{q-1}$ is changed to $2\sqrt{q-1}+1$) together with a
relaxed condition, the bound $2297$ is now $2351$.

######  Proposition 2.11.

Suppose that $V$ and $G$ satisfy the assumptions of the previous paragraph. If
$n(G,V)<2\sqrt{q-1}$, then $n=2$, $59\leq q=p\leq 2351$, and $|G/Z|\leq 24$.

###  2.5. Bounding $n(G,V)$

The following is Theorem 6.1 of [Mar16] with the following changes. The
occurrence of $59$ is changed to $53$, the lower bound $2\sqrt{q-1}$ is
changed to $2\sqrt{q-1}+1$, and $2297$ is changed to $2351$.

######  Theorem 2.12.

Let $V$ be a finite, faithful, coprime and irreducible $FG$-module. Suppose
that the characteristic $p$ of the underlying field $F$ is at least $53$. Put
$q=|F|$ and $|V|=q^{n}$. Let the center of ${\mathrm{GL}}(n,q)$ be $Z$. Then
$n(G,V)\geq 2\sqrt{q-1}+1$ unless possibly if one of the following cases
holds.

  1. (i)

$(G,V)\in\mathcal{C}_{q}$;

  2. (ii)

$V=\mathrm{Ind}_{H}^{G}(W)$ for some $2$-dimensional $FH$-module $W$ where $H$
is as $G$ in Proposition 2.10 or Proposition 2.11 satisfying one of the
following.

    1. (a)

$53\leq q\leq 14389$, $q\equiv\pm 1\pmod{10}$, and $2.\mathfrak{A}_{5}\leq
H/{\bf C}_{H}(W)\leq Z\circ 2.\mathfrak{A}_{5}$;

    2. (b)

$53\leq q=p\leq 2351$ and $|(H/{\bf C}_{H}(W))/{\bf Z}(H/{\bf C}_{H}(W))|\leq
24$.

Part of the proof of Theorem 2.12 is Lemma 2.13, which extends Lemma 2.8.

######  Lemma 2.13.

Let $(G,V)$ be a pair among the exceptions in Theorem 2.12, satisfying
$n(G,V)<2\sqrt{q-1}+1$. Then $|G|<|V|^{3/2}$.

###  2.6. Bounding $k(G)$

We now also have to take $k(G)$ into account.

######  Theorem 2.14.

Let $V$ be an irreducible and faithful $FG$-module for some finite group $G$
and finite field $F$ of characteristic $p$ at least $53$. Suppose that $p$
does not divide $|G|$. Then we have at least one of the following.

  1. (i)

$n(G,V)\geq 2\sqrt{p-1}+1$.

  2. (ii)

$k(G)\geq 2\sqrt{p-1}+1$.

  3. (iii)

$|V|=|F|=p$.

  4. (iv)

Case (2/a) of Theorem 2.12 holds with $p=59$ and $1<t\leq 15$, or $p=61$ and
$t=1$, or $61\leq p\leq 119$ and $2\leq t\leq 5$.

  5. (v)

Case (2/b) of Theorem 2.12 holds with $t\leq 4$.

Theorem 2.14 is Theorem 7.1 in [Mar16] with the following changes. One
occurrence of $59$ is now $53$, in conclusions (1) and (2) the bound
$2\sqrt{p-1}$ is now $2\sqrt{p-1}+1$, there is a minor change in conclusion
(3), and in conclusion (4) the $14$ became $15$ and the $4$ is now $5$. There
are minor changes in the proof of Theorem 2.14 with respect to the proof of
Theorem 7.1 of [Mar16].

###  2.7. Bounding $n(G,V)$ and $k(G)$

Minor changes were made in the corresponding section of [Mar16] to complete
the proof of Theorem 2.2, using Theorem 2.14 and a theorem of Hering [HB82,
Chapter XII].

##  3\. Known Results on principal blocks

In the following we will denote by $B_{0}(G)$ the principal $p$-block of the
group $G$. Recall that

${1}_{G}\in{\rm Irr}(B_{0}(G))=\\{\chi\in{\rm Irr}(G)\ |\ \sum_{{x\in
G}\atop{(p,o(x))=1}}\chi(x)\neq 0\\}\,.$

We collect useful facts about characters in principal $p$-blocks.

######  Lemma 3.1.

Let $G$ be a finite group, let $N$ be a normal subgroup of $G$ and let $p$ be
a prime.

  1. (i)

${\rm Irr}(B_{0}(G/N))\subseteq{\rm Irr}(B_{0}(G))$.

  2. (ii)

If $N$ is a group of order coprime to $p$, then ${\rm Irr}(B_{0}(G/N))={\rm
Irr}(B_{0}(G))$.

  3. (iii)

For any $\theta\in{\rm Irr}(B_{0}(N))$, there is some $\chi\in{\rm
Irr}(B_{0}(G))$ lying over $\theta$.

  4. (iv)

If $B_{0}(G)$ is the only block covering $B_{0}(N)$, then for any
$\theta\in{\rm Irr}(B_{0}(N))$ every constituent of $\theta^{G}$ lies in
$B_{0}(G)$. This happens, for instance, if $G/N$ is a $p$-group or if $P{\bf
C}_{G}(P)\subseteq N$ for some $P\in{\rm Syl}_{p}(G)$.

  5. (v)

For any $\chi\in{\rm Irr}(B_{0}(G))$, every constituent of $\chi_{N}$ lies in
$B_{0}(N)$.

###### Proof.

Part (i) follows from the definition of block domination [Nav98, p. 198], (ii)
is [Nav98, Theorem 9.9(c)], (iii) is [Nav98, Theorem 9.4], (iv) is [Nav98,
Theorem 9.6] and [RSV21, Lemma 1.3], and (v) follows from [Nav98, Theorem 9.2
and Corollary 9.3] by noticing that $B_{0}(N)$ is always $G$-invariant. ∎

######  Lemma 3.2.

Assume $G=H_{1}\times\dots\times H_{t}$. Then

$\operatorname{Irr}(B_{0}(G))=\\{\theta_{1}\times\dots\times\theta_{t}\mid\theta_{i}\in\operatorname{Irr}(B_{0}(H_{i}))\\}.$

###### Proof.

This follows directly from the definition of principal block. ∎

######  Lemma 3.3 (Alperin–Dade).

Let $N\lhd G$ and $p$ be a prime. Assume that $G/N$ is a group of order
coprime to $p$ and that $G=N{\bf C}_{G}(P)$ for some $P\in{\rm Syl}_{p}(N)$.
Then restriction yields a bijection between ${\rm Irr}(B_{0}(G))$ and ${\rm
Irr}(B_{0}(N))$.

###### Proof.

This is proved by Alperin [Alp76] when $G/N$ is solvable, and by Dade [Dad77]
in general. ∎

We also collect some results concerning the action of Galois automorphisms of
$p$-power order on principal blocks. If $P$ is a $p$-group, then observe that
a linear character $\lambda\in\operatorname{Lin}(P)$ is $\sigma$-invariant if
and only if it has order $p$. This happens if and only if
$\Phi(P)\subseteq{\rm ker}(\lambda)$.

######  Lemma 3.4.

Let $G$ be a finite group of order divisible by $p$ and assume $G$ has a
normal Sylow $p$-subgroup $P$. Then
$\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(G))=\operatorname{Irr}({G/{\bf
O}_{p^{\prime}}(G)\Phi(P)})$.

###### Proof.

This is [RSV20, Lemma 2.2(a)]. ∎

For a finite group $G$, let $\tau$ be an automorphism of the cyclotomic field
${\mathbb{Q}}_{|G|}$ and assume that $\tau$ fixes all $p^{\prime}$-roots of
unity in ${\mathbb{Q}}_{|G|}$ and that $\tau$ has $p$-power order. If $p=2$,
the $p$-power order condition is superfluous as the group ${\rm
Gal}(\mathbb{Q}_{|G|}/\mathbb{Q}_{|G|_{2^{\prime}}})$ is a 2-group. If $p$ is
odd, then $\tau\in\langle\sigma\rangle$, where $\sigma$ is as defined in the
Introduction.

######  Lemma 3.5.

Let $p$ be a prime and let $N\lhd G$. Assume that $G/N$ is a group of order
coprime to $p$. If $\chi\in{\rm Irr}(G)$ and $\theta\in{\rm Irr}(N)$ are such
that $[\chi_{N},\theta]\neq 0$, then $\chi$ is $\tau$-fixed if, and only if,
$\theta$ is $\tau$-fixed

###### Proof.

Let $G_{\theta}$ be the inertia group of $\theta$ and $\psi\in{\rm
Irr}(G_{\theta}|\theta)$ be the Clifford correspondent of $\chi$. If
$\chi^{\tau}=\chi$, then $\tau|_{\mathbb{Q}(\psi)}\in{\rm
Gal}({\mathbb{Q}}(\psi)/{\mathbb{Q}}(\chi))$ has $p$-power order. By [NT21,
Lemma 2.1(ii)], $\tau|_{{\mathbb{Q}}(\psi)}=1$ so $\psi^{\tau}=\psi$ and
consequently $\theta^{\tau}=\theta$. The reciprocal implication follows by
direct application of [NT19, Lemma 5.1]. ∎

######  Corollary 3.6.

Let $N\lhd G$ and $p$ be a prime. Assume that $G/N$ is a group of order
coprime to $p$.

  1. (i)

If $\chi\in{\rm Irr}(B_{0}(G))$ is $\tau$-fixed, then every constituent
$\theta\in{\rm Irr}(B_{0}(N))$ of $\chi_{N}$ is $\tau$-fixed.

  2. (ii)

If $\theta\in{\rm Irr}(B_{0}(N))$ is $\tau$-fixed then some constituent
$\chi\in{\rm Irr}(B_{0}(G))$ of $\theta^{G}$ is $\tau$-fixed.

Moreover, with the above notation, $\theta$ has degree coprime to $p$ if, and
only if, $\chi$ has degree coprime to $p$.

###### Proof.

This follows directly from Lemmas 3.5 and 3.1, together with the fact that
$\chi(1)/\theta(1)$ divides $|G/N|$ for any $\chi$ and $\theta$ such that
$[\chi_{N},\theta]\neq 0$ by [Nav18, Theorem 5.12]. ∎

The following is [RSV20, Lemma 2.4]. We give a proof here for completeness.

######  Lemma 3.7.

Let $p$ be a prime and let $H\leq G$. Assume that $p\nmid|G:H|$ and ${\bf
C}_{G}(Q)\subseteq H$ where $Q\in\operatorname{Syl}_{p}(H)$. Let
$\theta\in\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(H))$. Then there is
some $\chi\in\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(G))$ contained in
$\theta^{G}$.

###### Proof.

Notice that $B_{0}(H)$ is admissible in the sense of [Nav98, p. 213], so
$B_{0}(H)^{G}$ is defined. By [Nav98, Theorem 6.7], we have that
$B_{0}(H)^{G}=B_{0}(G)$. Write $B_{0}=B_{0}(G)$ and

$(\theta^{G})_{B_{0}}=\sum_{\chi\in\operatorname{Irr}_{p^{\prime}}(B_{0})}a_{\chi}\chi+\sum_{\psi\in\operatorname{Irr}(B_{0})\atop
p\mid\psi(1)}a_{\psi}\psi\,,$

where $a_{\chi}=[\chi,\theta^{G}]$. As $p\nmid|G:H|\theta(1)=\theta^{G}(1)$,
[Nav98, Corollary 6.4] implies that

$\sum_{\chi\in\operatorname{Irr}_{p^{\prime}}(B_{0})}a_{\chi}\chi(1)\not\equiv
0\bmod\,p.$

Now let $\mathcal{O}_{1},\dots,\mathcal{O}_{t}$ be the
$\langle\sigma\rangle$-orbits on the set of characters in
$\operatorname{Irr}_{p^{\prime}}(B_{0})$ lying over $\theta$. Since
$\theta^{G}$ is $\sigma$-invariant, we have that if two characters
$\chi,\eta\in\operatorname{Irr}_{p^{\prime}}(B_{0})$ lying over $\theta$ are
$\langle\sigma\rangle$-conjugate then $a_{\chi}=a_{\eta}$. Thus if
$\chi_{i}\in\mathcal{O}_{i}$ we have

$\sum_{\chi\in\operatorname{Irr}_{p^{\prime}}(B_{0})}a_{\chi}\chi(1)=\sum_{i=1}^{t}|\mathcal{O}_{i}|a_{\chi_{i}}\chi_{i}(1)$

is not divisible by $p$. Since $\langle\sigma\rangle$ is a $p$-group, this
forces one of $\mathcal{O}_{i}$ to have size $1$, as desired. ∎

Finally, we give an alternative proof of [RSV20, Lemma 2.5].

######  Lemma 3.8.

Let $N\lhd G$ and $p$ be a prime. Assume $N\cong S^{t}$ is a minimal normal
subgroup of $G$, where $S$ is a simple nonabelian group, and that $G/N$ is a
$p$-group. Assume there is some $X$-invariant
$\alpha\in\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(S))$ for
$X/S\in\operatorname{Syl}_{p}(\operatorname{Aut}(S)/S)$. Then there is some
$\chi\in\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(G))$ not containing $N$
in its kernel.

###### Proof.

Let $\mathcal{X}$ be the $\operatorname{Aut}(S)$-orbit of $\alpha$ and
$\mathcal{Y}=\\{\alpha_{1}\times\cdots\times\alpha_{t}\mid\alpha_{i}\in\mathcal{X}\\}\subseteq\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(N))$,
so that $|\mathcal{Y}|=|\mathcal{X}|^{t}$ is not divisible by $p$. Since $N$
stabilizes every element of $\mathcal{Y}$ and $G/N$ is a $p$-group, there is
some $G$-invariant $\lambda\in\mathcal{Y}$. Since $N$ is perfect, the
determinantal order $o(\lambda)$ is $1$. By [Nav18, Corollaries 6.2 and 6.4]
there is an extension
$\hat{\lambda}\in\operatorname{Irr}_{p^{\prime},\sigma}(G)$ of $\lambda$.
Since $G/N$ is a $p$-group,
$\hat{\lambda}\in\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(G))$ by Lemma
3.1(iv). ∎

##  4\. Simple groups

The aim of this section is to prove parts (i) and (ii) from Theorem B in the
Introduction. Throughout, let $\mathcal{G}$ denote the absolute Galois group
${\rm Gal}({\mathbb{Q}}^{\rm ab}/{\mathbb{Q}})$ and given a prime $p$, let
$\sigma\in\mathcal{G}$ be the Galois automorphism defined in the Introduction.
We begin with the following observation about sporadic and alternating groups.

######  Lemma 4.1.

Let $p$ be an odd prime, and let $S$ be an alternating group, sporadic simple
group, or the Tits group ${}^{2}\\!\operatorname{F}_{4}(2)^{\prime}$. Assume
$(p,S)\neq(3,J_{3})$. Then every $\chi\in{\rm Irr}(S)$ is almost $p$-rational.

###### Proof.

For the sporadic, Tits, and alternating groups $\mathfrak{A}_{n}$ with $n\leq
6$, we can see this directly from the Character Table Library in GAP [GAP22].
Let $S=\mathfrak{A}_{n}$ with $n\geq 7$ and let $\chi$ be an irreducible
character of ${\rm Aut}(S)=\mathfrak{S}_{n}$, whose characters take only
rational values. Then $\chi|_{S}$ contains at most two irreducible
constituents, which must be acted on by $\sigma$. Since $\sigma$ has $p$-power
order, it follows that $\sigma$ acts trivially on these constituents, proving
the claim. ∎

Much of the work will be centered around groups of Lie type, so we begin by
fixing some notation in that situation.

###  4.1. Notation for groups of Lie type

Before continuing, we set some notation for the remainder of this section,
surrounding simple groups of Lie type. So for now, let $S$ be a simple group
of Lie type. We will exclude the Tits group
${}^{2}\\!\operatorname{F}_{4}(2)^{\prime}$ from this list, instead treating
it separately with the sporadic groups.

We have $A:={\rm Aut}(S)=\widetilde{S}\rtimes D$, where $\widetilde{S}$
denotes the set of inner-diagonal automorphisms and $D$ is an appropriate
group of graph-field automorphisms (see, e.g. [GLS98, Theorems 2.5.12 and
2.5.14]). Note that $\widetilde{S}/S$ is abelian, and that $D$ is abelian
unless $S=\operatorname{D}_{4}(q)$, in which case $D=\mathfrak{S}_{3}\times C$
for a cyclic group $C$.

Let $\mathbf{G}$ be a simple, simply connected algebraic group over
$\bar{\mathbb{F}}_{q_{0}}$ for some prime $q_{0}$, and let
$F\colon\mathbf{G}\rightarrow\mathbf{G}$ be a Steinberg endomorphism such that
$S=G/{\bf Z}(G)$ with $G=\mathbf{G}^{F}$. Let
$\mathbb{G}:=\mathbf{G}_{\mathrm{ad}}$ be a simple algebraic group of adjoint
type, of the same type as $\mathbf{G}$. Then
$S\cong[\mathbb{G}^{F},\mathbb{G}^{F}]$ and $\widetilde{S}\cong\mathbb{G}^{F}$
(see [MT11, Prop. 24.21] and discussion after). We will make these
identifications.

The following is an application of the work of Tiep–Zalesski [TZ04] and will
be useful in the case of defining characteristic.

######  Lemma 4.2.

Let $S$ be as above, with $p=q_{0}\geq 3$. Assume that
$S\neq{}^{2}\\!\operatorname{B}_{2}(q^{2})$. Then every $\chi\in{\rm Irr}(S)$
is $\sigma$-invariant.

###### Proof.

This follows from [TZ04, Thm. 1.3 and Prop. 10.12]. ∎

Now, let $(\mathbf{G}^{\ast},F)$ and $(\mathbb{G}^{\ast},F)$ be dual to
$(\mathbf{G},F)$ and $(\mathbb{G},F)$, respectively. For each conjugacy class
$(s)$ of semisimple elements (that is, $q_{0}^{\prime}$-elements) in
$(\mathbb{G}^{\ast})^{F}$, there is a unique _semisimple_ character
$\chi_{s}\in{\rm Irr}(\widetilde{S})$, whose degree is
$[(\mathbb{G}^{\ast})^{F},{\bf
C}_{\mathbb{G}^{\ast}}(s)^{F}]_{q_{0}^{\prime}}$. These characters will play
an important role in our proof.

The unipotent characters will also play an important role. Thanks to the work
of Lusztig [Lus88], the unipotent characters of $S$ can simultaneously be
viewed as the deflation of unipotent characters of $G$ or as the irreducible
restriction of unipotent characters of $\widetilde{S}$. In particular, the
principal series unipotent characters are those that are constituents of the
induced character ${\mathrm{Ind}}_{B}^{G}(1_{B})$, where $B=\mathbf{B}^{F}$ is
the set of fixed points of an $F$-stable Borel subgroup $\mathbf{B}$ of
$\mathbf{G}$. By a slight abuse of terminology, if $\chi\in{\rm Irr}(S)$ is
the deflation of a principal series unipotent character of $G$, we will refer
to $\chi$ itself as a principal series unipotent character.

###  4.2. Part (i) of Theorem B

Although not completely necessary for our purposes here, we prove a stronger
version of part (i) of Theorem B in the case of groups of Lie type when $p\neq
q_{0}$. We hope that this can be useful in future work. This statement will
utilize the principal series unipotent characters.

######  Lemma 4.3.

Let $S=G/{\bf Z}(G)$ be a simple group of Lie type, where $G=\mathbf{G}^{F}$
is of simply connected type and $G\neq{}^{2}\\!\operatorname{F}_{4}(q)$. Let
$\chi\in\operatorname{Irr}(S)$ be a principal series unipotent character. If
$S\leq A\leq{\rm Aut}(S)$ is such that $A$ is generated by inner-diagonal and
field automorphisms, then there is an extension $\hat{\chi}$ of $\chi$ to $A$
that is $\mathcal{G}_{\chi}$-invariant.

###### Proof.

First, by [Mal08, Thms. 2.4 and 2.5], every unipotent character extends to its
stabilizer in ${\rm Aut}(S)$, which includes the inner-diagonal and field
automorphisms. If $F$ is a Frobenius morphism, then the statement follows
directly from [RS22, Prop. 2.6] and its generalization [Joh22, Prop. 8.7].
Otherwise, $G={}^{2}\\!\operatorname{G}_{2}(q)$ or
${}^{2}\\!\operatorname{B}_{2}(q)$, and $\chi$ is either the trivial character
or Steinberg character, so the result follows from [Sch92] in this case. ∎

######  Lemma 4.4.

Let $p$ be a prime and let $S=G/{\bf Z}(G)$ be a simple group of Lie type,
where $G=\mathbf{G}^{F}$ is of simply connected type defined in characteristic
distinct from $p$. Then ${\rm Irr}_{p^{\prime}}(B_{0}(S))$ contains a
nontrivial rational-valued principal series unipotent character $\chi$ such
that if $S\leq A\leq{\rm Aut}(S)$ is an almost simple group generated by
inner-diagonal and field automorphisms, then there is a rational-valued
extension of $\chi$ to $A$.

###### Proof.

If $p\in\\{2,3\\}$, then the Steinberg character $\mathrm{St}_{S}$ lies in
${\rm Irr}_{p^{\prime}}(B_{0}(S))$. Similarly, if $p\geq 5$ and $S$ is
${\mathrm{Sp}}_{4}(q)$ with $q$ even, $\operatorname{PSL}_{2}(q)$,
$\operatorname{PSL}_{3}^{\epsilon}(q)$, $\operatorname{D}_{4}(q)$,
${}^{3}\\!\operatorname{D}_{4}(q)$, or a Suzuki or Ree group, then, as in the
proof of [GRSS20, Props. 4.4 and 4.5], the Steinberg character
$\mathrm{St}_{S}$ lies in ${\rm Irr}_{p^{\prime}}(B_{0}(S))$. Further,
$\mathrm{St}_{S}$ is rational-valued and extends to a rational-valued
character of ${\rm Aut}(S)$, by [Sch92].

From here, we may assume that $p\geq 5$ and $F$ is a Frobenius endomorphism,
so that the extension property will follow from Lemma 4.3 once we have shown
that ${\rm Irr}_{p^{\prime}}(B_{0}(S))$ contains a rational-valued principal
series unipotent character.

We next suppose that $S$ is of exceptional type $\operatorname{G}_{2}(q)$,
$\operatorname{F}_{4}(q)$, $\operatorname{E}_{6}(q)$,
${}^{2}\\!\operatorname{E}_{6}(q)$, $\operatorname{E}_{7}(q)$, or
$\operatorname{E}_{8}(q)$. Here the principal series unipotent characters are
rational except for (in the notation of [Car93, Sec. 13]) the characters
$\phi_{512,11},\phi_{512,12}$ of $S=\operatorname{E}_{7}(q)$ and
$\phi_{4096,11},\phi_{4096,12},\phi_{4096,},\phi_{4096,}$ of
$\operatorname{E}_{8}(q)$, by [BC72, Thm. 2.9] and [Cur75]. Let $d$ be the
order of $q$ modulo $p$. If $d$ is a regular number, then $B_{0}(S)$ again
contains $\mathrm{St}_{S}$ (see, e.g. [RSV21, Lem. 3.6], which follows
directly from [Mal07, Thm. 6.6] and [Eng00, Theorem A]). Otherwise, we see
using the decompositions of $d$-Harish-Chandra series in [BMM93, Table 2] and
the fact that all characters of such a series lie in the same block by [Eng00,
Theorem A] to see that there is a rational-valued principal series unipotent
character in ${\rm Irr}_{p^{\prime}}(B_{0}(S))$.

Finally, assume $G$ is one of the remaining cases of classical type. In these
cases, every unipotent character is rational-valued. Here two unipotent
members of ${\rm Irr}_{p^{\prime}}(B_{0}(S))$ are illustrated in the proof of
[GRSS20, Prop. 4.4], given explicitly in Tables 1-4 of loc. cit. It suffices
to know that at least one of these characters lies in the principal series.
For $S=\operatorname{PSL}_{n}(q)$, every unipotent character lies in the
principal series. In the remaining cases, [Car93, Sec. 13.8] describes how to
determine, from the combinatorial description of the unipotent character,
whether it lies in the principal series. From this, we see that, indeed, at
least one of the two characters described in [GRSS20, Tables 1-4] lies in the
principal series in each case. ∎

With this, we are prepared to prove part (i) of Theorem B.

###### Proof of Theorem B(i).

If $S$ is a sporadic, Tits, or alternating group, then every character is
almost $p$-rational by Lemma 4.1. Then since $p\nmid|{\mathrm{Out}}(S)|$, it
suffices in these cases to note that $B_{0}(S)$ contains a nontrivial
$p^{\prime}$-degree character.

Next, suppose that $S$ is a simple group of Lie type defined in characteristic
$p$. We may find a simple, simply connected algebraic group $\mathbf{G}$
defined over $\overline{\mathbb{F}}_{p}$ such that $S=G/{\bf Z}(G)$, where
$G=\mathbf{G}^{F}$ for some Steinberg endomorphism $F$. Here again we have
every character is almost $p$-rational by Lemma 4.2. In [GRSS20, Props. 4.2
and 4.3] and building off of [GRS20, Prop. 4.3], a nontrivial member of ${\rm
Irr}_{p^{\prime}}(B_{0}(S))$ is exhibited that extends to $X$.

Finally, assume that $S$ is a simple group of Lie type defined in
characteristic distinct from $p$. Then since $p\geq 5$, we have $X$ such that
$X/S\in\operatorname{Syl}_{p}(A/S)$ can be chosen to be generated by inner-
diagonal and field automorphisms, so the statement follows from Lemma 4.4. ∎

###  4.3. Part (ii) of Theorem B

We will now turn our attention toward part (ii) of Theorem B. In many cases,
we will be able to use the following simplified condition:

######  Corollary 4.5.

Let $S\leq T$ be as in Theorem B(ii). Then Theorem B(ii) holds for $T$ if
there exist at least $2\sqrt{p-1}+1$ characters in
$\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(S))$ that are not $T$-conjugate.

###### Proof.

This follows from Corollary 3.6, and Clifford theory, as non-$T$-conjugate
members of ${\rm Irr}(S)$ cannot lie under the same character in ${\rm
Irr}(T)$. ∎

Lemma 4.2 will continue to be useful in defining characteristic. In the case
of non-defining characteristic, we will use unipotent characters and certain
semisimple characters to obtain the required bounds.

######  Lemma 4.6.

Let $p$ be a prime and assume that $p\neq q_{0}$. Let $\widetilde{\chi}\in{\rm
Irr}(\widetilde{S})$ be semisimple corresponding to a semisimple class
containing $s\in\mathbb{G}^{\ast}$ such $|s|=p$. Then
$\widetilde{\chi}\in\operatorname{Irr}_{\sigma}(B_{0}(\widetilde{S}))$.

Further, if $p\geq 5$ and either $S\neq\operatorname{PSL}_{n}^{\epsilon}(q)$
or $p\nmid(q-\epsilon)$, then $\widetilde{\chi}$ restricts irreducibly to $S$
and is not $D$-conjugate to $\widetilde{\chi}\beta$ for any nontrivial
$\beta\in{\rm Irr}(\widetilde{S}/S)$.

###### Proof.

Since $|s|=p$, we have the semisimple character $\chi_{s}\in{\rm
Irr}(\widetilde{S})$ corresponding to $s$ is fixed under $\sigma$, by e.g.
[ST18, Lem. 3.4]. Further, $\chi_{s}$ lies in $B_{0}(\widetilde{S})$ by
[His90, Cor. 3.4], so
$\chi_{s}\in\operatorname{Irr}_{\sigma}(B_{0}(\widetilde{S}))$.

Now, assume $p\geq 5$ and $S\neq\operatorname{PSL}_{n}^{\epsilon}(q)$ or
$p\nmid(q-\epsilon)$ and suppose that $\chi_{s}^{\varphi}=\chi_{s}\beta$ for
some $\varphi\in D$ (with the possibility $\varphi=1$) and $1\neq\beta\in{\rm
Irr}(\widetilde{S}/S)$. Then $\chi_{s}^{\varphi}=\chi_{sz}$ for some $1\neq
z\in{\bf Z}((\mathbb{G}^{\ast})^{F})$. Our conditions on $p$ ensure that
$p\nmid|z|$ for any such $z$, and hence $|s|\neq|sz|$. But we have
$s^{\varphi^{\ast}}$ is conjugate to $sz$ for some automorphism
$\varphi^{\ast}$ dual to $\varphi$ by [Tay18, Prop. 7.2], contradicting that
$|s|\neq|sz|$. We therefore obtain that no such $\varphi,\beta$ exist, and
hence we see $\chi_{s}$ is irreducible on restriction to $S$ (since the number
of consitutents in the restriction is the number of $\beta$ such that
$\chi_{s}=\chi_{s}\beta$), and $\chi_{s}$ is not $D$-conjugate to
$\chi_{s}\beta$ with $1\neq\beta$. ∎

We will also use the fact that unipotent characters are almost $p$-rational:

######  Lemma 4.7.

Let $p\geq 3$ be a prime distinct from $q_{0}$. Then any unipotent character
of $S$ or $\widetilde{S}$ is invariant under $\sigma$.

###### Proof.

For an integer $m$, let $\zeta_{m}$ denote a primitive $m$th root of unity.
First, note that $\sqrt{q}$ lies in ${\mathbb{Q}}(\zeta_{q_{0}},i)$, which is
fixed under $\sigma$ since $p\neq q_{0}$ and $p\geq 3$. Further, $\sigma$
fixes $\zeta_{r}$ for any prime $r$. Then from [Gec03, Table 1 and Prop. 5.6],
we see that the statement holds. ∎

With this, we are ready to prove Theorem B(ii).

###### Proof of Theorem B(ii).

If $S$ is sporadic, alternating $\mathfrak{A}_{n}$ with $n\leq 8$, Tits, or a
group of Lie type with nonexceptional Schur multiplier (see [GLS98, Tab.
6.1.3]), this can be checked in GAP [GAP22]. If $S=\mathfrak{A}_{n}$ with
$n\geq 9$, then the result follows from [HS23, Section 3] and the “if”
direction of Brauer’s Height Zero conjecture [KM13] if a Sylow $p$-subgroup of
$S$ is abelian, and by [HSV23, Proposition 3.1(iii)] otherwise, taking into
account Lemma 4.1.

We now let $S$ be a simple group of Lie type, defined in characteristic
$q_{0}$. Let $P\in\operatorname{Syl}_{p}(S)$ and assume $P$ is non-cyclic.
Note that since ${}^{2}\\!\operatorname{B}_{2}(q^{2})$ and
${}^{2}\\!\operatorname{G}_{2}(q^{2})$ have cyclic Sylow $p$-subgroups for
$p\geq 5$ (see, e.g. [GLS98, Theorem 4.10.2]), we may assume throughout that
$S$ is not one of these.

In most cases, we aim to exhibit at least $2\sqrt{p-1}+1$ characters in
$\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}({S}))$ that are not ${\rm
Aut}(S)$-conjugate and apply Corollary 4.5. Often, we will use the results of
[HS23, HSV23], together with Lemmas 4.2, 4.7, and 4.6. It will be useful to
note throughout that $p\geq 2\sqrt{p-1}+1$ for $p\geq 5$ and that
$x+\frac{p-1}{x}+1\geq 2\sqrt{p-1}+1$ for $p\geq 5$ and $x\geq 1$.

We first assume that $q_{0}\neq p$.

###### Non-Defining Characteristic with $P$ Abelian.

Let $P\in\operatorname{Syl}_{p}(S)$ be abelian. Note that by the “if”
direction of BHZ [KM13], we have ${\rm Irr}_{p^{\prime}}(B_{0}(S))={\rm
Irr}(B_{0}(S))$, and similar for $T$.

With this, the proof in this case follows almost directly from the arguments
in the proof of [HS23, Theorem 1.2], with some minor modification. In loc.
cit, the strategy is to exhibit enough unipotent characters and semisimple
characters $\chi_{s}$ with $s\in G^{\ast}$ a $p$-element that lie in
$B_{0}(S)$ and are not ${\rm Aut}(S)$-conjugate. However, we see that by
replacing arbitrary $p$-elements $s$ with elements only of order $p$, the
bounds obtained there are still sufficient, and now the characters lie in
$\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(S))$ thanks to Lemmas 4.7 and
4.6. We summarize the details. Throughout, we will let $\mathscr{U}$ denote
the set of unipotent characters in $B_{0}(S)$ and $\mathscr{S}$ the set of
characters in $B_{0}(S)$ that are the irreducible restrictions of semisimple
characters $\chi_{s}\in{\rm Irr}(\widetilde{S})$ with $s$ of order $p$. (That
is, $\mathscr{U}\cup\mathscr{S}$ is the set of characters for which we will be
able to apply Lemmas 4.7 and 4.6.)

Let $e$ denote the order of $q$ modulo $p$ and let $\Phi_{e}$ be the $e$th
cyclotomic polynomial in $q$ if $S\neq{}^{2}\\!\operatorname{F}_{4}(q^{2})$.
If $S={}^{2}\\!\operatorname{F}_{4}(q^{2})$, then let $\Phi_{e}$ instead be
the polynomial $\Phi^{(p)}$ over $\\\ Q(\sqrt{2})$ defined in [Mal07, Sec.
8.1]. Let $p^{a}$ be the largest power of $p$ dividing $\Phi_{e}$, and let
$\Phi_{e}^{k}$ be the order of a Sylow $\Phi_{e}$-torus of $(\mathbb{G},F)$.
Then a Sylow $p$-subgroup $\widetilde{P}$ of $\widetilde{S}$ and of
$(\mathbb{G}^{\ast})^{F}$ is isomorphic to $\widetilde{P}\cong C_{p^{a}}^{k}$.
The analogue of [HS23, Lemma 5.3] holds for elements $t\in\mathbb{G}^{\ast}$
of order $p$, yielding that the corresponding semisimple character $\chi_{t}$
of $\widetilde{S}$ has an orbit of size at most $p-1$ under a field
automorphism $\alpha$. With this, the considerations in [HS23, 5D] yield the
same bound as [HS23, (5.3)]. Namely, the number of ${\rm Aut}(S)$-orbits on
$\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(S))$ from semisimple characters
is at least $\frac{p^{k}-1}{dg(p-1)|W_{e}|}$, where $W_{e}$ is the so-called
relative Weyl group for a minimal $e$-split Levi subgroup of $(\mathbb{G},F)$,
$d=|\widetilde{S}/S|$, and $g$ is the size of the group of graph automorphisms
in $D$. Note that $k\geq 2$, since by assumption the Sylow $p$-subgroups of
$S$ are non-cyclic.

From here, the proof in [HS23, Theorem 8.1] gives Theorem B(ii)(b) when $S$ is
an exceptional group of Lie type. To obtain part (a), we again follow the
proof of [HS23, Theorem 8.1], where the methods there in fact yield our
required $\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(T))\geq 2\sqrt{p-1}+1$,
although only $2\sqrt{p-1}$ was required in [HS23].

Now suppose that $S$ is of classical type. First let
$S=\operatorname{PSL}_{n}^{\epsilon}(q)$ with $n\geq 3$. (Note that
$\operatorname{PSL}_{2}(q)$ has cyclic Sylow $p$-subgroups.) Let $e^{\prime}$
be the order of $\epsilon q$ modulo $p$, and let
$w=\lfloor\frac{n}{e^{\prime}}\rfloor$. Since $n\neq 2$, note that we have
$(e^{\prime},w)\neq(1,2)$. Here the proof of [HS23, Prop. 6.1] yields the
result. There we see that the number of ${\rm Aut}(S)$-orbits in $\mathscr{U}$
is at least $2e^{\prime}+1$ and that the number of ${\rm Aut}(S)$-orbits in
$\mathscr{S}$ least $\frac{p-1}{2e^{\prime}}$. Then we have at least
$2e^{\prime}+\frac{p-1}{2e^{\prime}}+1\geq 2\sqrt{p-1}+1$ ${\rm
Aut}(S)$-orbits in $\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(S))$, using
Lemmas 4.7 and 4.6.

Next, we let $S=\operatorname{C}_{n}(q)$ with $n\geq 2$,
$\operatorname{B}_{n}(q)$ with $n\geq 3$, $\operatorname{D}_{n}(q)$ with
$n\geq 4$, or ${}^{2}\\!\operatorname{D}_{n}(q)$ with $n\geq 4$. Here let
$e^{\prime}$ denote the order of $q^{2}$ modulo $p$ and again let
$w=\lfloor\frac{n}{e^{\prime}}\rfloor\geq 2$. Then the proof of [HS23, Prop.
7.1] shows that the number of ${\rm Aut}(S)$-orbits in $\mathscr{U}$ is at
least $4e^{\prime}$, and is strictly greater unless
$S\in\\{\operatorname{D}_{n}(q),{}^{2}\\!\operatorname{D}_{n}(q)\\}$ with
$(e^{\prime},w)=(2,2)$ or $S=\operatorname{C}_{2}(2^{f})$.

If $S=\operatorname{C}_{n}(q)$ or $\operatorname{B}_{n}(q)$, the proof in loc.
cit. then tells us that there are at least
$4e^{\prime}+\frac{p-1}{4e^{\prime}}+1\geq 2\sqrt{p-1}+1$ orbits under ${\rm
Aut}(S)$ of characters in $\mathscr{U}\cup\mathscr{S}$, unless
$S=\operatorname{C}_{2}(2^{f})$. In the latter case, we instead have at least
$4+\frac{p-1}{8}\geq 2(p-1)^{1/4}$ such orbits, giving (b). However, the proof
there also shows that the number of characters in $B_{0}(T)$ lying above
members of $\mathscr{U}\cup\mathscr{S}$ is at least $3b+\frac{(p-1)^{2}}{8b}$,
where $b=|T/S{\bf C}_{T}(P)|$. Since this number is at least $p$ for $p\geq 5$
and $b\geq 1$, we are again done in this case.

Now let $S=\operatorname{D}_{n}(q)$ with $n\geq 5$ or
$S={}^{2}\\!\operatorname{D}_{n}(q)$ with $n\geq 4$. In this case, the proof
of [HS23, Prop. 7.1] gives at least $4e^{\prime}+\frac{p-1}{4e^{\prime}}+1\geq
2\sqrt{p-1}+1$ orbits under ${\rm Aut}(S)$ of characters in
$\mathscr{U}\cup\mathscr{S}$ unless $e^{\prime}=2$. Suppose we are then in the
case $e^{\prime}=2$. Let $\Gamma$ be the group
$\widetilde{S}\leq\Gamma\leq{\rm Aut}(S)$ generated by inner, diagonal, and
graph automorphisms, and consider the group $X:=(\Gamma\cap T){\bf C}_{T}(P)$.
Then the Alperin–Dade correspondence Lemma 3.3 gives a bijection, via
restriction, between ${\rm Irr}(B_{0}(X))$ and ${\rm Irr}(B_{0}(\Gamma\cap
T))$. Further, $B_{0}(T)$ is the unique block covering $B_{0}(X)$ by Lemma
3.1(iv). Since there are at least $8$ ${\rm Aut}(S)$-orbits of characters in
$\mathscr{U}$, we have at least $8$ characters in
$\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(X))$ lying over members of
$\mathscr{U}$, using Lemma 4.7 and Corollary 3.6. Letting $b:=|T/X|$, this
gives at least $8b$ characters in
$\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(T))$ lying above these, since
the unipotent characters are fixed by field automorphisms and extend to their
inertia groups in ${\rm Aut}(S)$ by [Mal08, Theorems 2.4 and 2.5]. Further,
there are at least $\left(\frac{p-1}{4}\right)^{2}$ characters in
$\mathscr{S}$ using the arguments of [HS23, Theorem 7.1]. This gives at least
$\frac{1}{8}\left(\frac{p-1}{4}\right)^{2}$ characters in
$\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(X))$, and
$\frac{1}{8b}\left(\frac{p-1}{4}\right)^{2}$ in
$\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(T))$, lying above these. Hence,
we obtain that $|\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(T))|\geq
8b+\frac{(p-1)^{2}}{128b}$, which is larger than $2\sqrt{p-1}+1$ for $p\geq 5$
and $b\geq 1$.

Finally, consider $S=\operatorname{D}_{4}(q)$. Here the methods of the
penultimate paragraph of [HS23, Prop. 7.1] yield at least
$4+\frac{p-1}{24}\geq 2(p-1)^{1/4}$ orbits under ${\rm Aut}(S)$ in
$\mathscr{U}\cup\mathscr{S}$, giving (b). For (a), the final paragraph of loc.
cit gives at least $4b+\frac{(p-1)^{2}}{96b}$ characters in $B_{0}(T)$ lying
above members of $\mathscr{U}\cup\mathscr{S}$, where $b=|T/X|$ with
$X=(\Gamma\cap T){\bf C}_{T}(P)$ and
$\Gamma=\widetilde{S}\rtimes\mathfrak{S}_{3}$ is the group generated by
$\widetilde{S}$ and the group of graph automorphisms. This number is at least
$2\sqrt{p-1}+1$ for $p\geq 5$ and $b\geq 2$ or for $p>23$ and $b\geq 1$. At
this point, we remark that this bound can be improved to
$4b+\frac{(p-1)^{2}}{48b}+\frac{(p-1)}{12b}$, since the original bound in
[HS23] only considered semisimple elements corresponding to elements
$(\lambda_{1},\lambda_{2},\cdots,\lambda_{k})$ of $C_{p}^{k}$ with at most
$k-2$ values of $i$ such that $\lambda_{i}=1$, and since the bound
$\frac{(p-1)^{2}}{16}$ for the number of orbits of semisimple characters was
from restriction from the group $\operatorname{GO}_{8}(q)$, which therefore
had already taken into account the action of an order-two graph automorphism.
This number is more than $\lceil 2\sqrt{p-1}\rceil$ unless $p=11$ with $b=1$,
in which case the order $3$-graph automorphism must act trivially on the
semisimple elements of order $11$ corresponding to elements as above with
$\lambda_{i}=1$ for at least $k-2$ values of $i$. Then since
$4+10^{2}/16+10/4>\lceil 2\sqrt{10}\rceil$, we are again done in this case. ∎

###### Non-Defining Characteristic with $P$ Nonabelian.

Let $P$ be nonabelian. Then if $S$ is of exceptional type, this means that
$p\in\\{5,7\\}$, and hence it suffices to know that there are at least 6
unipotent characters in ${\rm Irr}_{p^{\prime}}(B_{0}(\widetilde{S}))$ that
are not $\operatorname{Aut}(S)$-conjugate, applying Lemma 4.7 and Corollary
4.5. The existence of these characters (recalling that
$P\in\operatorname{Syl}_{p}(S)$ is noncyclic and $p\geq 5$) is proven in
[RSV21, Lem. 3.7], except possibly if
$S=\widetilde{S}=\operatorname{G}_{2}(q)$ and $p\mid(q^{2}-1)$. In the latter
situation, loc. cit. provides at least $5$ non-${\rm Aut}(S)$-conjugate
unipotent characters in ${\rm Irr}_{p^{\prime}}(B_{0}(\widetilde{S}))$, and
taking any $s\in\mathbb{G}^{\ast}$ of order $p$, we obtain a semisimple
character $\chi_{s}$ that cannot be conjugate to any unipotent character, and
we are done using Lemma 4.6.

We now assume that $S$ is of classical type. In this case, parts (IV) and (VI)
of the proof of [HSV23, Prop. 3.2] shows that there are at least $p$ unipotent
characters in ${\rm Irr}_{p^{\prime}}(B_{0}(S))$ that are not
$\operatorname{Aut}(S)$-conjugate, and we are again done by applying Lemma
4.7.

∎

Finally, suppose that $q_{0}=p$.

###### Defining Characteristic.

Using Lemma 4.2 and Corollary 4.5, we are almost done using [HSV23, Prop.
3.2]. However, note that we must exibit possibly one additional character to
what was needed in loc. cit. Using part (II) of that proof, we have
$|\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(S))|\geq q^{r}/d$, where $r$ is
the semisimple rank of $G$ and $d=|\widetilde{S}/S|$ is the size of the
diagonal automorphism group. Then the number of ${\rm Aut}(S)$-orbits on
$\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(S))$ is at least
$\frac{q^{r}}{d\cdot|{\mathrm{Out}}(S)|}=\frac{q^{r}}{d^{2}\cdot f\cdot
g}\geq\frac{q^{r/2}}{d^{2}\cdot g}$, where $q=p^{f}$ and $g$ is the size of
the group of graph automorphisms, noting that
$q^{r}/f=p^{rf}/f\geq\sqrt{q^{r}}$. Now, from the knowledge of $d$ and $g$
from, e.g., [GLS98, Thm. 2.5.12], we have $\frac{q^{r}}{d^{2}\cdot f\cdot
g}\geq p$, except possibly if $S=\operatorname{PSL}_{4}^{\epsilon}(5)$;
$S=\operatorname{PSL}_{2}(q)$ with $q\in\\{p,25,49\\}$; or
$\operatorname{PSL}_{3}^{\epsilon}(p)$. Note that $\operatorname{PSL}_{2}(p)$
has a cyclic Sylow $p$-subgroup. In the remaining outlying cases, we have
$\frac{q^{r}}{d^{2}\cdot f\cdot g}\geq 2\sqrt{p-1}+1$, except possibly for
$\operatorname{PSL}_{4}^{\epsilon}(5)$,
$\operatorname{PSL}_{3}^{\epsilon}(5)$,
$\operatorname{PSL}_{3}^{\epsilon}(7)$,
$\operatorname{PSL}_{3}^{\epsilon}(11)$, and $\operatorname{PSL}_{2}(25)$. In
these cases, we see directly from their character tables in GAP that there are
at least $1+2\sqrt{p-1}$ orbits under ${\rm Aut}(S)$ in
$\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(S))$. ∎

This completes the proof of Theorem B(ii). ∎

##  5\. Cyclic Sylow $p$-subgroups

The aim of this section is to prove part (iii) from Theorem B of the
Introduction. We make use of the fact that simple nonabelian groups with
cyclic Sylow $p$-subgroups satisfy the inductive Alperin–McKay conditions (as
defined in [Spä13]) for odd primes [KS16], together with deep properties of
block character triple isomorphisms from [NS14]. We use the notation
$\sim_{b}$ from [NS14, Definition 3.6].

Assume $S$ has a cyclic Sylow $p$-subgroup $P$ with $p$ an odd prime. Then
following [KS16, Section 5] the characters in $B_{0}=B_{0}(S)$ can be written
as a disjoint union

$\operatorname{Irr}_{\mathrm{ex}}(B_{0})\cup\operatorname{Irr}_{\mathrm{nex}}(B_{0})$

where $\operatorname{Irr}_{\mathrm{nex}}(B_{0})$ is exactly the set of
$p$-rational characters of $B_{0}$ by [KS16, Lemma 5.5]. In particular,
$1_{S}\in\operatorname{Irr}_{\mathrm{nex}}(B_{0})$. If $\Lambda$ is a ${\bf
N}_{G}(P)$-transversal on $\operatorname{Irr}(P)\setminus\\{1_{P}\\}$ then for
$\lambda\in\Lambda$, we define

$\eta_{\lambda}=\sum_{g\in[{\bf N}_{G}(P)/{\bf C}_{G}(P)]}\lambda^{g},$

where $[{\bf N}_{G}(P)/{\bf C}_{G}(P)]$ is a transversal of ${\bf C}_{G}(P)$
in ${\bf N}_{G}(P)$. Notice that $\eta_{\lambda}$ does not depend on the
representative $\lambda\in\Lambda$.

Fixing some $\chi\in\operatorname{Irr}_{\mathrm{nex}}(B_{0})$, we have a
parametrization
$\operatorname{Irr}_{\mathrm{ex}}(B_{0})=\\{\chi_{\lambda}\mid\lambda\in\Lambda\\}$
where, by [KS16, Lemma 5.6] $\chi_{\lambda}$ is the unique non-$p$-rational
constituent of the generalized character

$\chi*\eta_{\lambda},$

defined in the main theorem of [BP80] ([Nav98, p. 114]). For principal blocks,
the $*$ construction can be simplified. If $x$ is a $p$-element, we denote by
$S(x)=\\{u\in G\mid u_{p}\text{ is $G$-conjugate to }x\\}$ (this is known as
the $p$-section of $x$ [Nav98, p. 105]).

######  Lemma 5.1.

Let $G$ be a finite group, $p$ a prime, $\chi\in\operatorname{Irr}(B_{0}(G))$
and $P\in{\rm Syl}_{p}(G)$. Let $\mathbb{S}$ be a $G$-transversal on the set
of $p$-elements of $G$ contained in $P$, then

$\chi*\eta_{\lambda}=\sum_{x\in\mathbb{S}}\eta_{\lambda}(x)\mathbf{1}_{S(x)}\chi$

where $\mathbf{1}_{S(x)}$ is the characteristic function of the $p$-section
$S(x)$ of $x$.

###### Proof.

Let $g\in G$, then $g\in S(x_{0})$ for a unique $x_{0}\in\mathbb{S}$. In
particular, $g^{u}=x_{0}y$ for some $u$ in $G$, for a unique
$x_{0}\in\mathbb{S}$ and some element $y\in{\bf C}_{G}(x_{0})$ of order
coprime to $p$. Then

$\chi*\eta(g)=\chi*\eta(xy)=\sum_{x\in{\mathbb{S}}}\eta(x)\chi^{(x,b_{0}(x))}$

where $b_{0}(x)=B_{0}({\bf C}_{G}(x))$.

Now as in [Nav98, p. 114], $\chi^{(x,b_{0}(x))}(g)=0$ if $g$ is not in $S(x)$
and if $g\in S(x)$ and $g^{u}=xy$ with $y\in{\bf C}_{G}(x)$ of order coprime
to $p$, we have

$\chi^{(x,b_{0}(x))}(g)=\sum_{\varphi\in{\rm
IBr}(b_{0}(x))}d^{x}_{\chi\varphi}\varphi(y)$

where $d^{x}_{\chi\varphi}$ are the generalized decomposition numbers [Nav98,
p. 100]. By Brauer’s Third Main theorem [Nav98, Theorem 6.7] and [Nav98,
Corollary 5.8] we have

$\sum_{\varphi\in{\rm
IBr}(b_{0}(x))}d^{x}_{\chi\varphi}\varphi(y)=\chi(xy)=\chi(g)$

and we are done. ∎

We can now easily check that the bijection constructed in [KS16, Section 6] is
equivariant with respect to the action of $\langle\sigma\rangle$ on
characters.

######  Lemma 5.2.

Let $p$ be an odd prime. Let $S$ be a finite group of order divisible by $p$.
Let $P\in{\rm Syl}_{p}(S)$. Suppose that $P$ is cyclic. Then the bijection

$\Omega:\operatorname{Irr}(B_{0}(S))\rightarrow\operatorname{Irr}(B_{0}({\bf
N}_{S}(P)))$

from [KS16, Section 6] is $(\langle\sigma\rangle\times{\bf N}_{{\rm
Aut}(S)}(P))$-equivariant. Moreover, every character in
$\operatorname{Irr}(B_{0}(S))$ is almost $p$-rational if, and only if,
$|P|=p$.

###### Proof.

Write $H={\bf N}_{S}(P)$, let
$\chi\in\operatorname{Irr}_{\mathrm{nex}}(B_{0}(S))$ and
$\Omega(\chi)=\chi^{\prime}\in\operatorname{Irr}_{\mathrm{nex}}(B_{0}(H))$. In
the bijection $\Omega$ from [KS16, Section 6],
$\operatorname{Irr}_{\mathrm{nex}}(B_{0}(S))$ maps onto
$\operatorname{Irr}_{\mathrm{nex}}(B_{0}(H))$ and the restriction
$\Omega:\operatorname{Irr}_{\mathrm{ex}}(B_{0}(S))\rightarrow\operatorname{Irr}_{\mathrm{ex}}(B_{0}(H))$
is defined by $\Omega(\chi_{\lambda})=\chi^{\prime}_{\lambda}$ with the
labeling from the beginning of this section. Since the $p$-rational characters
of $\operatorname{Irr}(B_{0}(S))$ are exactly the characters in
$\operatorname{Irr}_{\mathrm{nex}}(B_{0}(S))$ it follows that to check
$\langle\sigma\rangle$-equivariance it suffices to check it for the
exceptional characters.

It is straightforward to check that
$\eta_{\lambda}^{\sigma}=\eta_{\lambda^{\sigma}}$. Using Lemma 5.1 and the
fact that $\chi\in\operatorname{Irr}_{\mathrm{nex}}(B_{0}(S))$ is $p$-rational
we obtain that
$\chi*(\eta_{\lambda})^{\sigma}=(\chi*\eta_{\lambda})^{\sigma}$. Similarly,
$\chi^{\prime}*(\eta_{\lambda})^{\sigma}=(\chi^{\prime}*\eta_{\lambda})^{\sigma}$.
Now by [KS16, Lemma 5.6] $\chi_{\lambda}$ is the unique non-$p$-rational
constituent of $\chi*\eta_{\lambda}$, so
$\Omega(\chi^{\sigma})=\Omega(\chi)^{\sigma}$, and the
$\langle\sigma\rangle$-equivariance of $\Omega$ is proven. Furthermore,
$\Omega$ is ${\bf N}_{\operatorname{Aut}(S)}(P)$-equivariant by [KS16,
Proposition 6.1]. The second part of the statement follows from the first one.
∎

######  Proposition 5.3.

Let $N\triangleleft G$ and let $p$ be a prime. Suppose that
$N=S_{1}\times\dots\times S_{t}$ where $S_{i}$ are nonabelian simple groups
with cyclic nontrivial Sylow $p$-subgroups. Let
$P\in\operatorname{Syl}_{p}(N)$. Then there is a
$\langle\sigma\rangle\times{\bf N}_{G}(P)$-equivariant bijection

$\Omega:{\rm Irr}(B_{0}(N))\rightarrow{\rm Irr}(B_{0}({\bf N}_{N}(P)))$

such that if $\theta\in{\rm Irr}(B_{0}(N))$ then

$(G_{\theta},N,\theta)\sim_{b}({\bf N}_{G}(P)_{\theta},{\bf
N}_{N}(P),\Omega(\theta)).$

###### Proof.

Notice that under our assumptions $p$ must be odd. Let
$A_{i}=\operatorname{Aut}(S_{i})$. Write $P=Q_{1}\times\dots\times Q_{t}$
where $Q_{i}\in\operatorname{Syl}_{p}(S_{i})$. By Lemma 5.2 there is a
$\langle\sigma\rangle\times{\bf N}_{A_{i}}(Q_{i})$-equivariant bijection

$\Omega_{S_{i}}:\operatorname{Irr}(B_{0}(S_{i}))\rightarrow\operatorname{Irr}(B_{0}({\bf
N}_{S_{i}}(Q_{i})))\,.$

By [KS16, Theorem 7.6] $B_{0}(S_{i})$ satisfies the inductive Alperin–McKay
conditions and the character bijection is exactly $\Omega_{S_{i}}$.

Using Lemma 3.2 the construction of [Spä13, Theorem 7.9] and [NS14, Theorem
6.3] we have that the map

$\Omega:{\rm Irr}(B_{0}(N))\rightarrow{\rm Irr}(B_{0}({\bf N}_{N}(P)))$

defined by

$\Omega(\eta_{1}\times\dots\times\eta_{t})=\Omega_{S_{1}}(\eta_{1})\times\dots\times\Omega_{S_{t}}(\eta_{t})$

is $\langle\sigma\rangle\times{\bf N}_{G}(P)$-equivariant and whenever
$\theta\in{\rm Irr}(B_{0}(N))$ we have that

$(G_{\theta},N,\theta)\sim_{b}({\bf N}_{G}(P)_{\theta},{\bf
N}_{N}(P),\Omega(\theta))\,,$

as wanted. ∎

We now show that the principal $p$-block of groups $G$ with a semisimple
normal subgroup $N$ of index coprime to $p$ and whose simple factors have
cyclic Sylow $p$-subgroups satisfies [IN02, Conjecture D]. If $S$ is a finite
group with cyclic Sylow $p$-subgroups, then ${\rm Irr}(B_{0}(S))$ consists
only of characters of $p^{\prime}$-degree by work of Dade. It follows from
Lemma 3.2 that if $N=S\times\dots\times S$ then ${\rm Irr}(B_{0}(N))$ also
contains only characters of $p^{\prime}$-degree. By Lemma 3.1(iv) and
Corollary 3.6 we conclude that, for groups $X$ as in Theorem B(iii), ${\rm
Irr}(B_{0}(X))$ contains only characters of $p^{\prime}$-degree.

As usual, if $N\triangleleft\,G$ and $\theta\in\operatorname{Irr}(N)$, we
denote by $\operatorname{Irr}(G|\theta)$ the set of characters $G$ that lie
over $\theta$. We denote
$\operatorname{Irr}(B_{0}(G)|\theta)=\operatorname{Irr}(B_{0}(G))\cap\operatorname{Irr}(G|\theta)$.
Notice that $\operatorname{Irr}(B_{0}(G)|\theta)$ is not empty if and only if
$\theta\in\operatorname{Irr}(B_{0}(N))$ by Lemma 3.1(iii) and (v).

######  Proposition 5.4.

Let $N\triangleleft G$ and let $p$ be a prime. Suppose that $N$ is semisimple,
its simple factors have nontrivial cyclic Sylow $p$-subgroups, and that
$p\nmid|G:N|$. Let $P\in{\rm Syl}_{p}(G)$. Let $\tau$ be an automorphism of
the cyclotomic field ${\mathbb{Q}}_{|G|}$ and assume that $\tau$ has $p$-power
order and that it fixes all $p^{\prime}$-roots of unity in
${\mathbb{Q}}_{|G|}$. Then $\tau$ fixes equal numbers of (height zero)
characters in ${\rm Irr}(B_{0}(G))$ and ${\rm Irr}(B_{0}({\bf N}_{G}(P)))$.

###### Proof.

Notice that under our assumptions $p$ must be odd. Then
$\tau\in\langle\sigma\rangle$. Write $M={\bf N}_{N}(P)$ and $H={\bf
N}_{G}(P)$, so that $G=NH$ and $M=N\cap H$. By Proposition 5.3 there is a
$H\times\langle\sigma\rangle$-equivariant bijection

$\Omega\colon\operatorname{Irr}(B_{0}(N))\rightarrow\operatorname{Irr}(B_{0}(M))$

such that

$(G_{\theta},N,\theta)\sim_{b}(H_{\theta},M,\Omega(\theta))$

for every $\theta\in\operatorname{Irr}(B_{0}(N))$. By the definition of
$\sim_{b}$, [Nav98, Definition 3.6] and Brauer’s Third Main theorem [Nav98,
Theorem 6.7], this implies that

$|\operatorname{Irr}(B_{0}(G_{\theta})|\theta)|=|\operatorname{Irr}(B_{0}(H_{\theta})|\Omega(\theta))|$

for every $\theta\in\operatorname{Irr}(B_{0}(N))$. Let $\Delta$ be a complete
set if representatives of the $H$-orbits on
$\operatorname{Irr}_{\tau}(B_{0}(N))$. Since $\tau\in\langle\sigma\rangle$, we
have that $\Omega(\Delta)$ is a complete set of representatives of the
$H$-orbits on $\operatorname{Irr}_{\tau}(B_{0}(M))$. Then, by Corollary 3.6,
we have the partitions

$\operatorname{Irr}_{\tau}(B_{0}(G))=\coprod_{\eta\in\Delta}\operatorname{Irr}(B_{0}(G)|\eta)$

and

$\operatorname{Irr}_{\tau}(B_{0}(H))=\coprod_{\eta\in\Delta}\operatorname{Irr}(B_{0}(H)|\Omega(\eta))\,.$

By the Fong–Reynolds correspondence [Nav98, Theorem 9.14] we obtain

$|\operatorname{Irr}(B_{0}(G)|\eta)|=|\operatorname{Irr}(B_{0}(G_{\eta})|\eta)|=|\operatorname{Irr}(B_{0}(H_{\eta})|\Omega(\eta))|=|\operatorname{Irr}(B_{0}(H)|\Omega(\eta))|\,$

as desired. ∎

The statement of Theorem B(iii) is now a corollary.

######  Corollary 5.5.

Let $S$ be a finite nonabelian simple group and let $p$ be a prime. Suppose
that $S$ has nontrivial cyclic Sylow $p$-subgroups. Write $N=S^{t}$ with
$t\geq 2$. Suppose that $N\triangleleft\,X$ with $p\nmid|X:N|$. Then

$k_{0,\sigma}(B_{0}(X))=k_{0,\sigma}(B_{0}({\bf N}_{X}(P)))>2\sqrt{p-1}$

where $P\in\operatorname{Syl}_{p}(X)$.

###### Proof.

Let $P\in{\rm Syl}_{p}(X)$. By Proposition 5.4 we have that

$k_{0,\sigma}(B_{0}(X))=k_{0,\sigma}(B_{0}({\bf N}_{X}(P))).$

Since ${\bf N}_{X}(P)$ has a normal Sylow $p$-subgroup, by Lemma 3.4, we have
that

$k_{0,\sigma}(B_{0}({\bf N}_{X}(P)))=k({\bf N}_{X}(P)/\Phi(P){\bf
O}_{p^{\prime}}({\bf N}_{X}(P)))\,.$

The fact that $P/\Phi(P)>1$ is not cyclic implies that

$k({\bf N}_{X}(P)/\Phi(P){\bf O}_{p^{\prime}}({\bf N}_{X}(P)))>\lceil
2\sqrt{p-1}\rceil\,$

by Theorem 2.1 and this concludes the proof. ∎

##  6\. Proof of Theorem A

We begin by proving the inequality part in Theorem A.

######  Theorem 6.1.

Let $p$ be a prime and $G$ be a finite group of order divisible by $p$. Then
$k_{0,\sigma}(B_{0}(G))\geq 2\sqrt{p-1}$.

###### Proof.

If $G$ has a normal Sylow $p$-subgroup $P$, then using Lemma 3.4 we have
$k_{0,\sigma}(B_{0}(G))=k(G/\Phi(P){\bf O}_{p^{\prime}}(G))\geq 2\sqrt{p-1}$
by the main result of [Mar16] and we are done in this case.

We proceed by induction on $|G|$. By Lemma 3.1(ii) and the inductive
hypothesis, we may assume that ${\bf O}_{p^{\prime}}(G)=1$. If
$M\triangleleft\,G$ and $p\mid|G:M|$ then by induction and Lemma 3.1(i) we
have $k_{0,\sigma}(B_{0}(G))\geq k_{0,\sigma}(B_{0}(G/M))\geq 2\sqrt{p-1}$ and
we are done. Thus we may assume that $G$ has a unique minimal normal subgroup
$N$, which is semisimple, of order divisible by $p$ and $p\nmid|G:N|$.

Say $N\cong S^{t}$, where $S$ is nonabelian simple of order divisible by $p$.
Suppose that $S$ has cyclic Sylow $p$-subgroups. If $t=1$, then the Sylow
$p$-subgroups of $G$ are also cyclic and we are done by Lemma 5.2. If $t\geq
2$ then we apply Theorem B(iii).

Therefore, we may assume that the Sylow $p$-subgroups of $S$ are not cyclic.
Let $k$ be the number of $\operatorname{Aut}(S)$-orbits on
$\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(S))$. If $t\geq 2$, since
$G\leq\operatorname{Aut}(S)\wr\mathfrak{S}_{t}$, then $G$ has at least
$\binom{k+t-1}{t}$ orbits on
$\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(N))$. By Theorem B(ii)(b) we
have that $k\geq 2(p-1)^{1/4}$. In particular $G$ has at least

$\frac{k(k+1)}{2}=2\sqrt{p-1}+(p-1)^{1/4}\geq 2\sqrt{p-1}+1>\lceil
2\sqrt{p-1}\rceil$

orbits on $\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(N))$. Since $|G:N|$ is
not divisible by $p$, every such orbit lies under a distinct
$\chi\in\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(G))$ so we are done using
Corollary 3.6. If $t=1$, then the result follows from Theorem B(ii)(a). ∎

######  Theorem 6.2.

Assume $k_{0,\sigma}(B_{0}(G))=\lceil 2\sqrt{p-1}\rceil$. Then the Sylow
$p$-subgroups of $G$ are cyclic.

###### Proof.

Let $G$ be a minimal counterexample to the statement. In particular $P\in{\rm
Syl}_{p}(G)$ is not cyclic.

Step 1: We may assume $p\geq 5$.

If $p\leq 3$, then $\lceil 2\sqrt{p-1}\rceil=p$, so
$k_{0,\sigma}(B_{0}(G))=\lceil 2\sqrt{p-1}\rceil$ implies that $P$ is cyclic
by the main result of [RSV20], contradicting the choice of $G$ as a minimal
counterexample.

Step 2: If $N$ is a minimal normal subgroup of $G$, then $p$ divides $|N|$ and
$p$ does not divide $|G:N|$.

Assume otherwise. If $p\nmid|N|$ then
$\operatorname{Irr}(B_{0}(G))=\operatorname{Irr}(B_{0}(G/N))$ by Lemma
3.1(ii). By the minimality of $G$ we have that $G/N$ has cyclic Sylow
$p$-subgroups, then so does $G$, a contradiction.

If $p$ divides $|G:N|$ then by Theorem 6.1 we have

$\lceil
2\sqrt{p-1}\rceil\leq\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(G/N))\leq\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(G))=\lceil
2\sqrt{p-1}\rceil.$

Thus $G/N$ has cyclic Sylow $p$-subgroups and every $p^{\prime}$-degree almost
$p$-rational character of $B_{0}(G)$ lies over $1_{N}$ (and in the principal
block of $G/N$).

Suppose that $N$ is a $p$-group. Let $P\in\operatorname{Syl}_{p}(G)$ and
notice that $P$ is not cyclic but $P/N$ is cyclic and nontrivial. We have
$N\cap\Phi(P)<\Phi(P)$. Let $\lambda\in\operatorname{Irr}(P/\Phi(P))$ be such
that $N$ is not contained in ${\rm ker}(\lambda)$. Then $\lambda$ is
$\sigma$-invariant and linear. Write $P{\bf C}_{G}(P)=P\times X$ and
$\hat{\lambda}=\lambda\times
1_{X}\in\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(P{\bf C}_{G}(P)))$. By
Lemma 3.7 there is some
$\chi\in\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(G))$ contained in
$\hat{\lambda}^{G}$. By Frobenius reciprocity, $\chi_{P{\bf C}_{G}(P)}$
contains $\hat{\lambda}$ so $N$ is not contained in ${\rm ker}(\chi)$, a
contradiction.

Therefore, we may assume that $N$ is a direct product of a nonabelian simple
group $S$ of order divisible by $p$. Write $N=S_{1}\times\dots\times S_{t}$,
where $S_{i}\cong S$ for every $i$. By Theorem 6.1 and the hypothesis that
$k_{0,\sigma}(B_{0}(G))=\lceil 2\sqrt{p-1}\rceil$, we have that

$k_{0,\sigma}(B_{0}(G/N))=\lceil 2\sqrt{p-1}\rceil$

so it suffices to show that there is some
$\chi\in\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(G))$ such that $N$ is not
contained in ${\rm ker}(\chi)$.

Write $M=PN$ and $H=M{\bf C}_{G}(P)$. By Theorem B(i) and Lemma 3.8 there is
some $\psi\in\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(M))$ not containing
$N$ in its kernel. By Alperin–Dade (Lemma 3.3), there is some extension
$\tilde{\psi}\in\operatorname{Irr}(B_{0}(H))$ of $\psi$, and by Lemma 3.5 we
have $\tilde{\psi}\in\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(H))$. By
Lemma 3.7, the induced character $(\tilde{\psi})^{G}$ contains some
$\chi\in\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(G))$, and since
$\chi_{N}$ contains $\theta$, $N$ is not contained in ${\rm ker}(\chi)$.

Final step. Let $N$ be a minimal normal subgroup of $G$. By Step 2 we have
$|G:N|$ is not divisible by $p$. Therefore, whenever
$\theta\in\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(N))$ there is some
$\chi\in\operatorname{Irr}_{p^{\prime},\sigma}(B_{0}(G))$ lying over $\theta$.
By Step 2 we have that $N$ is semisimple, and $N$ is the only minimal normal
subgroup of $G$. If $N=S_{1}\times\dots\times S_{t}$ with $t\geq 2$, then
arguing as in Theorem 6.1 we obtain that $k_{0,\sigma}(B_{0}(G))>\lceil
2\sqrt{p-1}\rceil$, so we may assume $N$ is simple and $G$ is almost simple.
If $N$ has cyclic Sylow $p$-subgroups, then so does $G$, contradicting the
choice of $G$ as a counterexample. Otherwise, Theorem B(ii) shows $G$ is not a
counterexample. ∎

##  7\. Final comments

###  7.1. Related problems

Fix a prime number $p$ and write

$\mathcal{S}_{p}=\\{e+\frac{p-1}{e}\ |\ e\text{ divides }p-1\\}.$

In [HMM22], Hung, Malle and the first-named author of this article ask the
following question.

######  Question 7.1 (Question 1.5 of [HMM22]).

Is it true that the Sylow $p$-subgroups of a group $G$ of order divisible by
$p$ are cyclic if $k_{0,\sigma}(B_{0}(G))\in\mathcal{S}_{p}$?

Notice that when $p-1$ is a perfect square, then $\lceil
2\sqrt{p-1}\rceil=2\sqrt{p-1}={\rm min}(\mathcal{S}_{p})$. Then our main
theorem provides a partial positive answer to the above question. It is still
unknown whether there are finitely many primes $p$ satisfying that $p-1$ is a
perfect square. In fact, this is one of the 4 problems on prime numbers that
Landau presented in his talk at the International Congress of Mathematicians
in 1912.

In [CK23] the authors put forward the following conjecture.

######  Conjecture 7.2 (Cınarcı–Keller).

Let $p$ be a prime. Let $a$ and $b$ be positive integers such that $p-1=ab$
and such that $|a-b|$ is minimal. Then, for any finite group $G$ of order
divisible by $p$, we have $k(G)\geq a+b$ and $k(G)=a+b$ if, and only if,
$G\cong\mathsf{C}_{p}\rtimes\mathsf{C}_{a}$ or
$\mathsf{C}_{p}\rtimes\mathsf{C}_{b}$.

It is straightforward to see that $a+b$ is precisely the minimum of
$\mathcal{S}_{p}$, where $\mathcal{S}_{p}$ is defined as above. Notice that
$\lceil 2\sqrt{p-1}\rceil\leq\min\mathcal{S}_{p}$, so our Theorem 2.1 provides
evidence for the Cınarcı–Keller conjecture in the case where $\lceil
2\sqrt{p-1}\rceil=\min\mathcal{S}_{p}$ or, equivalently, when $\lceil
2\sqrt{p-1}\rceil\in\mathcal{S}_{p}$. This happens significantly more often
than $\sqrt{p-1}\in\mathbb{Z}$ (a quick check in [GAP22] shows that $\lceil
2\sqrt{p-1}\rceil\in\mathcal{S}_{p}$ for $77$ out of the $168$ primes smaller
than $1000$, whereas there are only $10$ primes smaller than $1000$ with
$\sqrt{p-1}\in\mathbb{Z}$).

###  7.2. Arbitrary blocks

Recall that the Héthelyi-Külshammer conjecture predicts that

$k(B)\geq 2\sqrt{p-1}$

whenever $B$ is a $p$-block with nontrivial defect of some finite group $G$.
The inequality holds for $p\leq 3$ by the weak block orthogonality relation.
As already mentioned in the Introduction, it was recently settled for
principal blocks in [HS23]. However, outside these cases, little is known
about this conjecture. In particular, it is not known whether it follows from
the Alperin-McKay conjecture nor whether it holds for solvable groups.

The following problem appeared recently in [Nav23].

######  Problem 7.3 (Problem 2.5 of [Nav23]).

Let $V$ be a finite dimensional $\mathbb{F}_{p}H$-module, where $H$ is a
finite $p^{\prime}$-group and $\mathbb{F}_{p}$ is the field of $p$ elements.
Let $G=VH$ be the semidirect product. Let $Z={\bf O}_{p^{\prime}}(G)={\bf
C}_{H}(V)$ and assume $Z\subseteq{\bf Z}(G)$. Let
$\lambda\in\operatorname{Irr}(Z)$. Then $|\operatorname{Irr}(G|\lambda)|\geq
2\sqrt{p-1}$.

As mentioned before its statement in [Nav23], if Problem 7.3 has a positive
answer, then the blocks $B$ satisfying the Alperin–McKay conjecture also
satisfy the Héthelyi–Külshammer conjecture. Recall that the Alperin-McKay
conjecture holds for a block $B$ if $k_{0}(B)=k_{0}(b)$, where $b$ is the
Brauer first main correspondent of $B$. If instead we assume that the block
$B$ satisfies the Alperin–McKay–Navarro conjecture [Nav04, Conjecture], then
$B$ would satisfy the blockwise version of our Theorem A. We provide a proof
of this fact in Propostion 7.5 below, but let us first recall the statement of
the Alperin–McKay–Navarro conjecture.

Let $\mathcal{H}\leq{\rm Gal}(\mathbb{Q}_{|G|}/\mathbb{Q})$ be the subgroup
generated by the field automorphisms $\tau$ which send $p^{\prime}$-roots of
unity $\xi$ to $\xi^{p^{e}}$ for some integer $e$. For a $p$-block $B$ we let
$\operatorname{Irr}_{0}(B)$ denote the set of characters of $B$ with height
zero. The group $\mathcal{H}$ acts on the set $\\{{\rm Irr}(B)\ |\ B\text{
$p$-block of $G$}\\}$, and we denote by $\mathcal{H}_{B}$ the stabilizer of
${\rm Irr}(B)$ under the action of $\mathcal{H}$.

######  Conjecture 7.4 (Alperin–McKay–Navarro).

Let $B$ be a $p$-block of $G$ with defect group $D$ and let $b$ be its Brauer
correspondent block in ${\bf N}_{G}(D)$. Then there is an
$\mathcal{H}_{B}$-equivariant bijection
$\operatorname{Irr}_{0}(B)\rightarrow\operatorname{Irr}_{0}(b)$.

Notice that our field automorphism $\sigma$ from the introduction is an
element of $\mathcal{H}$. Moreover the action of $\langle\sigma\rangle$
stabilizes ${\rm Irr}(B)$ for every $p$-block $B$ (because Brauer characters
are $\sigma$-invariant). In particular, Conjecture 7.4 predicts that the
action of $\langle\sigma\rangle$ on characters fixes the same number of height
zero characters in $B$ as it does in its Brauer correspondent block $b$, that
is

$k_{0,\sigma}(B)=k_{0,\sigma}(b)\,.$

######  Proposition 7.5.

Let $G$ be a finite group of order divisible by $p$, $B$ a $p$-block of $G$
with nontrivial defect group $D$. Assume Problem 7.3 has a positive answer and
that the Alperin–McKay–Navarro conjecture holds for $B$. Then

$k_{0,\sigma}(B)\geq 2\sqrt{p-1}\,.$

###### Proof.

Let $b$ be the Brauer correspondent block of $B$ in ${\bf N}_{G}(D)$. We have
that $k_{0,\sigma}(B)=k_{0,\sigma}(b)$ by the Alperin–McKay–Navarro
conjecture. By [VR23, Lemma 2.2] we have that

$k_{0,\sigma}(b)=k(\overline{b})$

where $\overline{b}$ is a block of ${\bf N}_{G}(D)/\Phi(D)$ with defect
$D/\Phi(D)$. By [Rey63, Theorem 6] applied to $\overline{b}$ there is a block
$b^{\prime}$ of a finite group $K$ with normal defect group
$V\in\operatorname{Syl}_{p}(K)$, $V\cong D/\Phi(D)$ and with a height-
preserving bijection
$\operatorname{Irr}(\overline{b})\rightarrow\operatorname{Irr}(b^{\prime})$.
It follows that $k_{0,\sigma}(B)=k(b^{\prime})$. By Schur–Zassenhaus, the
group $V$ has a complement $H$ in $K$, so $K=VH$ with $V$ an
$\mathbb{F}_{p}H$-module. By Fong’s theorem [Nav98, Theorem 10.20], there is
some $K$-invariant $\lambda\in\operatorname{Irr}({\bf O}_{p^{\prime}}(K))$
with $\operatorname{Irr}(b^{\prime})=\operatorname{Irr}(K|\lambda)$. By
character triple isomorphisms [Nav98, Problem 8.13] we may assume ${\bf
O}_{p^{\prime}}(K)\subseteq{\bf Z}(K)$. Using the assumption that Problem 7.3
has a positive answer, we have

$k_{0,\sigma}(B)=k(b^{\prime})=|\operatorname{Irr}(K|\lambda)|\geq
2\sqrt{p-1}$

as desired. ∎

The version for principal blocks of Problem 7.3 is the case where $Z=1$, and a
positive answer in this case is guaranteed by [Mar16]. Arguing as above, we
get that if the Alperin–McKay–Navarro conjecture holds for $B_{0}(G)$ then

$k_{0,\sigma}(B_{0}(G))=k({\bf N}_{G}(P)/\Phi(P){\bf O}_{p^{\prime}}({\bf
N}_{G}(P)))$

and the equality part of Theorem A follows from Theorem 2.1.

We close our note with a last observation. Notice that for every group $G$ and
every prime $p$ we have that $k_{0,\sigma}(B_{0}(G))\geq 1$ because the
principal $p$-block contains the trivial character $1_{G}$. However, the
existence of almost $p$-rational characters of height zero in arbitrary
$p$-blocks had not been observed until quite recently (see [NR24]). Now that
we know that $k_{0,\sigma}(B)\geq 1$ for arbitrary blocks and primes, we can
guarantee that the lower bound

$k_{0,\sigma}(B)\geq 2\sqrt{p-1}$

holds whenever $p\leq 3$ by a direct application of [RSV20, Lemma 1.4].

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(A. Maróti) Alfréd Rényi Institute of Mathematics, H-1053 Budapest, Hungary

Email address<EMAIL_ADDRESS>
(J. M. Martínez) Dipartimento di Matematica e Informatica ‘Ulisse Dini’, 50134
Firenze, Italy

Email address<EMAIL_ADDRESS>
(A. A. Schaeffer Fry)  (primary) Dept. Mathematics - University of Denver,
Denver, CO 80210, USA; and Dept. Mathematics and Statistics - MSU Denver,
Denver, CO 80217, USA

Email address<EMAIL_ADDRESS>
(C. Vallejo) Dipartimento di Matematica e Informatica ‘Ulisse Dini’, 50134
Firenze, Italy

Email address<EMAIL_ADDRESS>