# Notes on finite totally $2$-closed permutation groups

Gang Chen  School of Mathematics and Statistics, and Hubei Key Laboratory of
Mathematical Sciences, Central China Normal University, P.O. Box 71010, Wuhan
430079, P. R. China<EMAIL_ADDRESS>and  Qing Ren  School of
Mathematics and Statistics, and Hubei Key Laboratory of Mathematical Sciences,
Central China Normal University, P.O. Box 71010, Wuhan 430079, P. R. China.
<EMAIL_ADDRESS>
###### Abstract.

Let $N$ be a normal subgroup of a finite group $G$. For a faithful $N$-set
$\Delta$, applying the university embedding theorem one can construct a
faithful $G$-set $\Omega$. In this short note, it is proved that if the
$2$-closure of $N$ in $\Omega$ is equal to $N$, then the $2$-closure of $N$ in
$\Delta$ is also equal to $N$; in addition, it is proved that any abelian
normal subgroup of a finite totally $2$-closed group is cyclic; finally, it is
proved that if a finite nilpotent group is a direct of two nilpotent subgroups
where the two factors have coprime orders and both of them are totally
$2$-closed then $G$ is totally $2$-closed. As corollaries, several well-known
results on finite totally $2$-closed groups are reproved in more simple ways.

Keywords: totally $2$-closed group, $2$-closure, normal abelian subgroup

MSC Classification: 20B05, 20D10, 20D25.

The authors are supported by the NSFC grant No. 11971189.

##  1\. Introduction

Let $G$ be a finite group acting faithfully on a finite set $\Omega$. For a
positive integer $k$, $G$ acts naturally on the Catesian product
$\Omega^{k}:=\Omega\times\ldots\times\Omega$. In the year 1969, Wielandt [10,
Definition 5.3] introduced the definition of $k$-closure $G^{(k),\Omega}$ of
$G\leq{\rm Sym}(\Omega)$, which is defined as the largest subgroup of ${\rm
Sym}(\Omega)$ leaving each orbit of $G$ on $\Omega^{k}$ invariant. A finite
group $G$ is said to be a totally $k$-closed group, if $G=G^{(k),\Omega}$ for
any faithful $G$-set $\Omega$.

In recent years, finite totally $k$-closed groups have been studied in several
papers. In [1], it is proved that the center of every finite totally
$2$-closed group is cyclic; and finite nilpotent totally $2$-closed groups are
characterized. In [5], for a finite abelian group $G$, the minimal positive
integer $k$ for which $G$ is totally $k$-closed is given. Actually, the
minimal positive integer $k$ is proved to be $1$ plus the number of invariant
factors of $G$. In [3], it is proved that finite soluble totally $2$-closed
groups must be nilpotent; the Fitting subgroup of a totally $2$-closed group
is shown to be totally $2$-closed. In [2], the nontrivial finite totally
$2$-closed groups with trivial Fitting subgroups are classified.

In this short note, we will focus on finite totally $2$-closed groups. Our
first result, Theorem A, is a property on $2$-closures, the second result,
Theorem B, is a property on finite totallly $2$-closed goups, and the third
result, Theorem C, is on construction of finite nilpotent totally $2$-closed
groups. Applying our main results, several results on finite totally
$2$-closed groups will be reproved in self-contained and more simple ways.

Let $G$ be a finite group and $N$ a normal subgroup of $G$. Assume $\Delta$ is
a faithful $N$-set. By university embedding theorem (ref. Theorem 3.1), we may
construct a faithful $G$-set $\Omega$. Our first result is the following:

Theorem A. Keep the above notation. If $N^{(2),\Omega}=N$, then
$N^{(2),\Delta}=N$.

Theorem A captures the common features of the Fitting subgroup, the
centralizer of any normal subgroup in a finite totally $2$-closed group. And
these subgroups are proved to be totally $2$-closed in a totally $2$-closed
group, see Corollary 3.1.

Base on Theorem A and the known result that every finite totally $2$-closed
abelian group is cyclic, we have the second main result of this paper.

Theorem B. Every normal abelian subgroup of a finite totally $2$-closed group
is cyclic.

Since the classification of finite $p$-groups in which every normal abelian
subgroup is cyclic is well known, applying this classification it is proved
that a finite nilpotent totally $2$-closed groups is either cyclic or a direct
product of a generalized quarternion goup with a cyclic group of odd order.

Additionally, based on the known result that the $2$-closure of any finite
nilpotent permutation group is still nilpotent we have the following result,
which is the third main result of the paper.

Theorem C. Let $G=H\times K$ be a finite nilpotent group with $(|H|,|K|)=1$
and both $H$ and $K$ totally $2$-closed. Then $G$ is totally $2$-closed.

By Theorem B and Theorem C, the result on the characterization of finite
nilpotent totally $2$-closed groups in [1] is reproved in a more simple way.

In the final section of this paper, we discuss some known results on
$2$-closures of permutation groups in contrast with some results on coherent
configurations .

Notation. Throughout the paper, $\Omega$ denotes a finite set and the
symmetric group of $\Omega$ is denoted by ${\rm Sym}(\Omega)$. The Cartesian
product $\Omega\times\Omega$ is written as $\Omega^{2}$.

All groups are assumed to be finite nontrivial groups; if a finite group $G$
acts on $\Omega$ faithfully, $\Omega$ is also called a faithful $G$-set.

If a group $G$ acts on $\Omega$, the image of $\alpha\in\Omega$ under the
action of $g\in G$ is denoted by $\alpha^{g}$.

The center of a group $G$ is denoted by $Z(G)$, the centralizer of a subgroup
$N$ is denoted by $C_{G}(N)$, and the Fitting subgroup of $G$ is denoted by
$F(G)$.

If $H$ is a subgroup of a group $G$, the core of $H$ in $G$ is dentoed by
$H_{G}$, i.e., $H_{G}$ is the intersection of all conjugates of $H$ in $G$.

For a prime $p$, the finite $p$-group $M_{p^{n+1}}$ is defined as

$M_{p^{n+1}}=\langle a,b|a^{p^{n}}=b^{p}=1,b^{-1}ab=a^{1+p^{n-1}}\rangle.$

The cyclic group of order $n$ is denoted by $C_{n}$.

##  2\. Preliminaries

Let $G$ be a finite group and $\Omega$ a faithful $G$-set. The $2$-closure of
$G$ is defined as the largest subgroup of ${\rm Sym}(\Omega)$ that leaves
every orbit of $G$ on $\Omega^{2}$ invariant.

######  Lemma 2.1.

([10, Theorem 5.6]) Let $\Omega$ be a faithful $G$-set. Then $x\in
G^{(2),\Omega}$ if and only if for any $\alpha,\beta\in\Omega$ there exists
$g\in G$ such that $(\alpha,\beta)^{x}=(\alpha,\beta)^{g}$.

Note that in the above lemma for a fixed element $x\in G^{(2),\Omega}$ the
element $g$ depends on the choice of the pair $(\alpha,\beta)\in\Omega^{2}$.
Also, it is easily seen that $G^{(2),\Omega}$ contains $G$ as a subgroup.

######  Lemma 2.2.

([1, Lemma 2.2])Let $G\leq{\rm Sym}(\Omega)$ and $H\leq{\rm Sym}(\Gamma)$, and
let $(\theta,\lambda)$ be a permutation isomorphism from $(G,\Omega)$ to
$(H,\Gamma)$. Then $(\theta,\lambda)$ can be extended a permutation
isomorphism $(\tilde{\theta},\tilde{\lambda})$ from $(G^{(2),\Omega},\Omega)$
to $(H^{(2),\Gamma},\Gamma)$, where $\tilde{\lambda}=\lambda$ and the
restriction of $\tilde{\theta}$ to $G$ is equal to $\theta$.

######  Lemma 2.3.

([1, Lemma 2.1], [3, Lemma 1.1]) Let $G$ be a finite totally $2$-closed group
and $N$ a subgroup of $G$. Then $C_{G}(N)^{(2),\Omega}=C_{G}(N),$ for any
finite faithful $G$-set $\Omega$.

The following lemma is a special case of Lemma 3.2 of [5]. For completeness,
we give a proof.

######  Lemma 2.4.

([5, Lemma 3.2])Let $m,n$ be positive integers with $m$ divisible by $n$. Then
$C_{n}\times C_{m}$ is not totally $2$-closed.

Proof. Let $\Omega=\\{1,2,\ldots,2n+m\\}$. Choose two permutations in ${\rm
Sym}(\Omega)$ in the following way:

$x_{1}=(1\ldots n)(n+1,\ldots,2n),x_{2}=(n,n-1,\ldots,1)(2n+1,\ldots,2n+m).$

One can easily see that the subgroup $H=\langle x_{1},x_{2}\rangle$ of ${\rm
Sym}(\Omega)$ is isomorphic to the direct product $C_{n}\times C_{m}$.
However, the permutation $(1\ldots n)$ belongs to $H^{(2),\Omega}$ but not to
$H$. The lemma then follows.

$\square$

######  Lemma 2.5.

([3, Lemma 2.1]) Let $G=HK$ be a totally 2-closed group, where $H$ and $K$ are
proper subgroups of $G$. If $H_{G}\cap K_{G}=1$, then $G=H_{G}\times K_{G}$.
Furthermore, $H_{G}$ and $K_{G}$ are totally 2-closed. In particular, if
$H\cap K=1$, then $G=H\times K$ and both $H$ and $K$ are totally 2-closed.

######  Lemma 2.6.

([1, Theorem 3])If $G$ is a finite totally $2$-closed abelian group, then $G$
is cyclic.

Proof. It suffices to show that for every prime divisor $p$ of $|G|$, the
Sylow $p$-subgroup $S_{p}$ of $G$ is cyclic.

Suppose on the contrary that the Sylow $p$-subgroup $S_{p}$ of $G$ is not
cyclic for some prime divisor $p$ of $|G|$. Then there exist positive integers
$e_{1}\geq e_{2}$ and a subgroup $K$ of $G$ such that

$G=(C_{p^{e_{1}}}\times C_{p^{e_{2}}})\times K.$

By Lemma 2.5, $C_{p^{e_{1}}}\times C_{p^{e_{2}}}$ must be totally $2$-closed.
This is a contradiction to Lemma 2.4. $\square$

######  Lemma 2.7.

([8, Chap. 3, Theorem 7.6])Let $p$ be a prime and $G$ a finite nontrivial
$p$-group such that every normal abelian subgroup of $G$ is cyclic. Then

  1. (i)

If $p>2$, then $G$ is cyclic.

  2. (ii)

If $p=2$, then $G$ has a cyclic subgroup of index $2$.

The following lemma is based on the fact that as permutation groups cyclic
groups of prime power order and generalized quaternion groups have trivial one
point stabilizers.

######  Lemma 2.8.

([1, Lemma 2.5]) Finite cyclic groups of prime power orders and generalized
quaternion groups are totally $2$-closed.

######  Lemma 2.9.

([3, Corollary 3.4]) Let $G\leq{\rm Sym}(\Omega)$. Then $G$ is nilpotent if
and only if $G^{(2),\Omega}$ is nilpotent.

##  3\. Main Results

The following result plays a key role in the whole theory.

######  Theorem 3.1.

(University embedding theorem [7, Theorem 2.6 A]) Let $G$ be an arbitrary
group with a normal subgroup $N$ and put $K:=G/N$. Let $\psi:G\rightarrow K$
be a homomorphism of $G$ onto $K$ with kernel $N$. Let $T:=\\{t_{u}|u\in K\\}$
be a set of right coset representatives of $N$in $G$ such that $\psi(t_{u})=u$
for each $u\in K$. Let $x\in G$ and $f_{x}:\,K\rightarrow N$ be the map with
$f_{x}(u)=t_{u}xt^{-1}_{u\psi(x)}$ for all $u\in K$. Then
$\phi(x)=:(f_{x},\psi(x))$ defines an embedding $\phi$ of $G$ into $N\wr K$.
Furthermore, if $N$ acts faithfully on a set $\Delta$ then $G$ acts faithfully
on $\Omega:=\Delta\times K$ by the following rule
$(\delta,k)^{x}=(\delta^{f_{x}(k)},k\psi(x))$.

In the first paragraph on page 9 of [3], it was shown that the action of $N$
on $\Omega$ is permutation isomorphic to the natural action of $N$on a
disjoint union of $|G:N|$ copies of $\Delta$. Here, for each $g\in K$, the
corresponding copy may be denoted as

$\Delta_{g}:=\\{(\delta,g):\delta\in\Delta\\},$

and the permutation isomorphism is given by

$(\theta_{g},\lambda_{g}):\,(N,\Delta_{g})\,\rightarrow\,(N,\Delta),\quad
n\mapsto t_{g}nt_{g}^{-1},\,(\delta,g)\mapsto\delta.$

Then $N$ acts on $\Omega$ in the following way: for any $\delta\in\Delta$,
$g\in K$, and $x\in N$,

$(\delta,g)^{x}=(\delta^{\theta_{g}(x)},g).$

One can obtain a permutation isomorphism
$(\mu_{g},\lambda_{g}):(N^{(2),\Delta_{g}},\Delta_{g})\rightarrow(N^{(2),\Delta},\Delta)$,
by Lemma 2.2, for each $g\in K$. Also, the restriction of $\mu_{g}$ to $N$ is
equal to $\theta_{g}$. Furthermore, $N^{(2),\Delta_{g}}$ acts faithfully on
$\Omega$ in the following way:

$(\delta,g)^{x}=(\delta^{\mu_{g}(x)},g),\quad\forall\delta\in\Delta,g\in
K,\quad\text{and}\quad x\in N^{(2),\Delta_{g}}.$

For any $x\in N^{(2),\Delta}$, denote the images of $x$ in ${\rm Sym}(\Omega)$
by $\tilde{x}$. For any $(\delta,g)\in\Omega$,

$(\delta,g)^{\tilde{x}}=(\delta,g)^{\mu_{g}^{-1}(x)}=(\delta^{x},g).$ (3.1)

The following theorem, which is Theorem A in this paper, is a generalization
of [3, Proposition 3.6].

######  Theorem 3.2.

In the context of Theorem 3.1 , if we keep the above notation and assume that
$N^{(2),\Omega}=N$, then $N^{(2),\Delta}=N$.

Proof. It suffices to show that, as permutation groups on $\Omega$, the
cardinality of the image of $N^{(2),\Delta}$ in ${\rm Sym}(\Omega)$ is not
greater than that of $N$.

For any $x\in N^{(2),\Delta}$, and any
$(\delta_{1},g_{1}),(\delta_{2},g_{2})\in\Omega$, there exists $n\in N$ such
that $(\delta_{1},\delta_{2})^{x}=(\delta_{1},\delta_{2})^{n}$. Denote the
images of $x$ and $n$ in ${\rm Sym}(\Omega)$ by $\tilde{x}$ and $\tilde{n}$,
respectively. By formula (3.1), one can see that

$\begin{split}((\delta_{1},g_{1}),(\delta_{2},g_{2}))^{\tilde{x}}&=((\delta_{1}^{x},g_{1}),(\delta_{2}^{x},g_{2}))\\\
&=((\delta_{1}^{n},g_{1}),(\delta_{2}^{n},g_{2}))\\\
&=((\delta_{1},g_{1}),(\delta_{2},g_{2}))^{\tilde{n}}.\end{split}$

We conclude that ${\tilde{x}}\in N^{(2),\Omega}=N$. Thus, the cardinality of
the image of $N^{(2),\Omega}$ is not greater than that of $N$, as required.

$\square$

######  Corollary 3.1.

([3, Theorem B, Proposition 3.7]) Let $G$ be a finite totally $2$-closed group
and $N$ a normal subgroup of $G$. Then $C_{G}(N)$, $Z(G)$, and $F(G)$ are
totally $2$-closed group.

Proof. The corollary is a consequence of Lemma 2.3, Theorem 3.2, and [3, Lemma
3.5].

$\square$

######  Lemma 3.1.

([1, Theorem 1])The center of every totally $2$-closed group is cyclic.

Proof. By Corollary 3.1, the center is totally $2$-closed and thus is cyclic
by Lemma 2.6. $\square$

The following result is the second main result, Theorem B, in this paper.

######  Theorem 3.3.

Let $G$ be a finite totally $2$-closed group and $N$ a normal abelian subgroup
of $G$. Then $N$ is cyclic.

Proof. By Corollary 3.1, $C_{G}(N)$ is totally $2$-closed. Thus the center of
$C_{G}(N)$ is cyclic by Lemma 3.1. Note that $N$ is a central subgroup of
$C_{G}(N)$. It then follows that $N$ is cyclic. $\square$

######  Theorem 3.4.

Let $p$ be a prime and $G$ is a finite totally $2$-closed nontrivial
$p$-group. Then either $G$ is cyclic or $G$ is a generalized quaternion group.

Proof. Assume that $G$ is a finite totally $2$-closed $p$-group and that $G$
is not cyclic.

By Theorem 3.3, every normal abelian subgroup of $G$ is cyclic. By Lemma 2.7,
we see that $p=2$ and $G$ has a cyclic subgroup of index $2$. By [4, Theorem
1.2], $G$ is either a dihedral group, a generalized quaternion group,
$M_{2^{n+1}}$, or a semidihedral group.

However, except for the generalized quaternion group, the other three groups
are semidirect products of two proper subgroups. By Lemma 2.5, if $G$ is equal
to a semidirect product of two proper subgroups, then $G$ must be the direct
product of these two proper subgroups, which is a contradiction. We thus
conclude that $G$ is a generalized quaternion group.

$\square$

As a consequence of Lemma 2.5 and Theorem 3.4, we obtain the following.

######  Corollary 3.2.

If $G$ is a finite nilpotent totally $2$-closed group, then $G$ is cyclic or a
direct product of a generalized quaternion group with a cyclic group of odd
order.

The following theorem is the third main result, Theorem C, in this paper.

######  Theorem 3.5.

Let $G=H\times K$ be a finite nilpotent group with $(|H|,|K|)=1$ and both $H$
and $K$ totally $2$-closed. Then $G$ is totally $2$-closed.

Proof. Let $\Omega$ be a faithful $G$-set. By Lemma 2.9, $G^{(2),\Omega}$ is
nilpotent. We may write

$G^{(2),\Omega}=J\times L,$

where $(|J|,|L|)=1$, $H\leq J$, and $K\leq L$.

Set $j:=|J|$ and $l:=|L|$. For any $x\in G^{(2),\Omega}$ and any
$(\alpha,\beta)\in\Omega^{2}$, there exists $g\in G$ such that
$(\alpha,\beta)^{x}=(\alpha,\beta)^{g}$ by Lemma 2.1. By the assumption, one
can write $x=yz$ with $y\in J$ and $z\in L$, and $g=hk$ with $h\in H$ and
$k\in K$. Thus,

$(\alpha,\beta)^{x^{l}}=(\alpha,\beta)^{g^{l}}\,\Rightarrow\,(\alpha,\beta)^{y^{l}}=(\alpha,\beta)^{h^{l}}.$

This implies that $y^{l}\in H^{(2),\Omega}$ by Lemma 2.1. As we are assuming
that $H$ is totally $2$-closed, $y^{l}\in H$. In addition, the fact that
$(l,|H|)=1$ yields that $y\in H$. Similarly, one can show that $z\in K$.

It follows that $x=yz\in HK=G$ and therefore $G^{(2),\Omega}=G$. Since this is
true for any faithful $G$-set, $G$ is totally $2$-closed.

$\square$

######  Corollary 3.3.

Let $G$ be a finite cyclic group or a direct of a generalized quaternion group
with a cyclic group of odd order. Then $G$ is totally $2$-closed.

Proof. By the assumption we may decompose $G$ into a direct product of its
Sylow subgroups, where each Sylow subgroup is totally $2$-closed by Lemma 2.7.
Thus, $G$ is totally $2$-closed by Theorem 3.5. $\square$

As consequences of Corollaries 3.2 and 3.3, we have the following.

######  Corollary 3.4.

([1, Theorem 2]) A finite nilpotent group is totally $2$-closed if and only if
it is cyclic or a direct product of a generalized quaternion group with a
cyclic group of odd order.

##  4\. Concluding Remarks

Let $G\leq{\rm Sym}(\Omega)$. Then the action of $G$ on $\Omega^{2}$ generates
a combinatorial structure, i.e., the coherent configuration associated with
$G$, denoted by $\operatorname{Inv}(G)$. The $2$-closure of $(G,\Omega)$ is
then the automorphism group of the coherent configuration. In other words,
$G^{(2),\Omega}=\operatorname{Aut}(\operatorname{Inv}(G))$, see [6, Definition
2.2.14].

Several known results on $2$-closures of finite permutation groups have their
correspondents in the theory of coherent configurations. For example, Theorem
3 in [3] corresponds to Theorem 3.2.21 and Theorem 3.4.6 in [6]. And Corollary
3.2 in [3] corresponds to Exercise 2.7.17 $(4)$ in [6]. Finally, Lemma 1.4 in
[3] or Lemma 2.9 in [1] is a consequence of Theorem 3.2.5 in [6]. More
precisely, if $G_{i}\leq{\rm Sym}(\Omega_{i})$ and $\Omega$ is the disjoint
union of $\Omega_{i}^{\prime}s$, $i=1,\ldots,n$, then
$G=G_{1}\times\cdots\times G_{n}$ acts faithfully on $\Omega$ and
$G^{(2),\Omega}=G_{1}^{(2),\Omega_{1}}\times\cdots\times
G_{n}^{(2),\Omega_{n}}$ by the cited Theorem.

Acknowledgements. The authors would like to thank the refree for valuable
comments.

## References

  * [1] A. Abdollahi and M.Arezoomand, _Finite nilpotent groups that coincide with their $2$-closures in all of their faithful permutation representations_. J. Algebra Appl. 17(4) (2018), 1850065.
  * [2] M.Arezoomand, M. A. Iranmanesh, C. E. Praeger and G.Tracey, _Totally $2$-closed finite groups with trivial Fitting subgroup_, arxiv preprint, arxiv:2111.02253, 2021.
  * [3] A. Abdollahi, M. Arezoomand, and G.Tracey, _On finite totally $2$-closed groups_, arxiv preprint, arxiv:2001.09597, 2021.
  * [4] Y. Berkovich, _Groups of prime power order,_ Vol. 1, Expositions in Mathematics 46, Walter de Gruyter GmbH $\&$ Co.KG, Berlin, 2008.
  * [5] D. Churikov and C. E. Praeger, _Finite totally $k$-closed groups_, Tr. Inst. Mat. Mekh. 27 (1)(2021), 240-246.
  * [6] G. Chen and I. Ponomarenko, _Coherent Configurations_ , Central China Normal Universit Press, Wuhan, 2019.
  * [7] J.D. Dixon and B. Mortimer, _Permutation groups_ , Springer, New York, 1996.
  * [8] B. Huppert, _Endliche Gruppen. I._ Die Grundlehren der Mathematischen Wissenschaften, Band 134. Spring-Verlag, Berlin-New York, 1967.
  * [9] H. Wielandt, _Finite permutation groups,_ Academic Press, New York, 1964.
  * [10] H.W. Wielandt, _‘Permutation groups through invariant relations and invariant functions’_ , Lecture Notes, Ohio State University, 1969. Also published in: Wielandt, Helmut, Mathematische Werke (Mathematical works) Vol. 1. Group theory, Walter der Gruyter $\&$ Co., Berlin, 1994, pp. 237-296.


David Alvarez-Castillo

# Simulation of the isotropic ultra-high energy photons flux in the solar
magnetic field and a comparison with observations made by the HAWC and Fermi-
LAT observatories

Piotr Homola, Bożena Poncyljusz, Dariusz Gora, Dhital Niraj

###### Abstract

In this contribution we study the possibility of the formation of cosmic ray
ensembles (CRE) created by the interaction of ultra-high energy (UHE) photons
with the magnetic field of the Sun. The lack of observation of those UHE and
the difficulties for their identification given the current methodologies
motivates this study. We performed simulations using the PRESHOWER program in
order to simulate the expected extensive air showers which might be spatially
correlated generated upon entering the Earth’s atmosphere. We found
characteristic features like very thing and extremely elongates cascades of
secondary photons with their corresponding energies spanning the entire cosmic
range spectrum. Shower footprints are as large as hundreds of kilometres. An
application of this study is the scenario of gamma-ray emission from the
vicinity of the Sun as a result of ultra-high energy photon cascading in the
solar magnetic field in order to understand recent observations made by the
HAWC and Fermi-LAT observatories.

##  1 Introduction

Ultra-high energy (UHE) photons expected to be produced by different
mechanisms like decays of super heavy dark matter particles, with their
expected flux being much higher than the observed one. One explanation for
this is the cosmic rays ensembles phenomenon that consists of the production
of one or more particle cascades created by the UHE photon in space that would
eventually reach the Earth, interact with its atmosphere resulting in multiple
showers extremely extended. This scenario is opposite to the one of a single
particle or photon interacting with the atmosphere and creating a single,
geographically localized shower.

Interestingly, the Fermi-Large Area Telescope (Fermi-LAT) reported the
detection of gamma-ray emission with energies 1-100 GeV (up to 200 GeV) from
the solar disk at the quiescent phase, while the Sun is the source of MeV-GeV
photons [1, 2]. Thus, interactions of cosmic rays with the solar atmosphere
were considered as the source of detected gamma-rays. However, the mechanism
of photons emission remains unknown. Moreover, further observations of the
solar photon flux in the TeV range did not provide any signal [3]. Therefore,
in our approach, we consider the interactions of ultra-high energy (UHE)
photons with the solar magnetic field as the cause of gamma-ray emission.
According to the works of [6, 4, 5], the UHE photon may convert to the
positron-electron pair in the magnetic field and then positron and electron
generate the secondary photon cascades via the magnetic Bremsstrahlung
process. The secondary photon cascade features a characteristic signature at
the top of the atmosphere, in the shape of a highly elongated ellipse. The
cascades produce the cosmic-ray ensembles (CRE) in the atmosphere - a group of
air showers correlated in time or space, which are impossible to detect by a
single observatory [7]. Furthermore, the dependence between the density of
photons produced in the cascading process nearby the Sun, and the solar
magnetic field variations in different phases of the solar activity cycle may
play an important role related to modulations [8, 9].

Previous results [6] showed that for the primary photon with energy 100 EeV,
aiming at the Earth’s centre, there is the maximum of the secondary photons
energy distribution for around 1 TeV. For lower energies of UHE photon the
corresponding maximum should be also shifted towards the GeV range of
energies. Furthermore, the magnetic pair production requires sufficiently high
magnetic field, so it is possible to occur only at the close vicinity of the
Sun. Then, secondary photons reaching the atmosphere have directions enclosed
in the solid angle defined by the solar disk and its halo. Therefore, UHE
photon cascading in the solar magnetic field is an interesting candidate for
the source of detected gamma-ray emission and it is significant to perform a
detailed study on the cumulative distribution of secondary photons on the top
of the atmosphere for the UHE photons energy range consistent with the current
limits on the observed UHE photon flux [10]. According to the previous study
[6] a possible anisotropy may be observed in the distribution of the
orientation of signatures at the top of the atmosphere. Therefore, it could be
a potential observable, which may enable the UHE photon cascading
identification.

##  2 Model of ultra-high energy photon interaction

The model of interaction of UHE photons in the vicinity of the Sun has already
been presented in [6, 11] where simulations in order to derive the
corresponding particles footprints were performed. Solar magnetic fields have
been described by either by a dipole configuration or by the so called dipole-
quadrupole-current-sheet (DQCS) model [12]. Simulations are carried out by
means of PRESHOWER program [6, 13] with modifications which generates
secondary photons on the top of the atmosphere. In order to collect a
satisfying amount of data, we optimized the existing PRESHOWER simulator. In
the new version of the program we firstly generate the landing point of the
UHE photon at the observation sphere in such a manner that the position
distribution is isotropic (but we consider only the half of the atmosphere
sphere which is seen from the Sun). The observation sphere centre is the Earth
centre and the radius is choose to be 104 km. Then, we isotropically generate
the random direction from the solid angle defined by the Sun’s vicinity (we
take 10 $R_{\odot}$ from the Sun’s centre) without the solar disc. The UHE
photon is propagated until it converts or reaches the atmosphere. If the
conversion occurs the predicted orientation of the secondary photons signature
is calculated. The predicted signature is given by the equation

$\theta=\arctan\left(\frac{t_{x}B_{y}-t_{y}B_{x}}{t_{z}B_{x}-t_{x}B_{z}}\right),$
(1)

where $t_{i}$ \- the UHE photon trajectory unit vector and $B_{i}$ \- the
magnetic field at the conversion point. The $z$ axis is determined by the
Earth and Sun centres (the origin of coordinate system is at the Earth’s
centre) and we choose the $x$ axis parallel to the solar equatorial plane, the
ecliptic. Fig. 1 shows that the predicted signature’s orientation is an
excellent approximation of the exact orientation.

Figure 1: Comparison of predicted and exact signatures’ orientation angle for
an exemplary UHE photon energy 3 EeV within the DQCS model of the solar
magnetic field.

Due to the computation time limitations it was considered only the start
sphere of 104 km radius, neglecting some potentially observable cases when the
UHE photon aims at more distant points. Furthermore, it not possible to
discuss the rate of observed cases for a different start sphere radius, since
the simulation was stopped at the conversion point. Therefore, we can only
refer to the discussion of neglected cases in the previous version of the
PRESHOWER. In that version we have considered UHE photons reaching the sphere
of 7 $R_{\odot}$ radius around the Sun, aiming at the atmosphere plane.

##  3 Results

Applied optimisation enabled us to create the cumulative distribution of
signatures’ orientation based on 10 000 events of CRE effects for UHE photons
energies: 1, 3, 10, 30 and 100 EeV and for the dipole and DQCS model of the
solar magnetic field. Obtained results are shown in the Fig. 2.

(a) 3 EeV

(b) 10 EeV

(c) 30 EeV

(d) 100 EeV

(e) 1 EeV

Figure 2: The distribution of signatures’ rotation for different UHE photon
energies.

It is clearly visible that the model of the solar magnetic fields used in
simulations greatly affects the signatures’ orientation distribution. The most
prominent anisotropies in the distribution may be observed for energies 1 and
100 EeV and for the DQCS model. However, 100 EeV UHE photons are strongly
suppressed by the present limits [10]. Moreover, for the UHE photons energy 1
EeV, which are much more frequently observed, the conversion occurs only for
the DQCS model and still very rarely. Therefore, we needed to compute the
ratios of conversion cases to all simulated cases and taking into account the
present limits on the UHE photons flux we calculated the realistic signatures’
orientation distribution for the whole range of UHE photons energies (1-100
EeV).

Figure 3: The calculated fraction of simulations giving CRE effects in all
executed simulations as a function of the UHE photon energy.

Calculated ratios of conversion cases to all simulated cases are shown in the
Figure 3. For the dipole model and the energy 1 EeV, the result should be
interpreted as the upper limit on conversion cases rate, since no conversion
occurred for the 50 000 of simulations. On the basis of [10], we adopted the
limits on the UHE photons flux presented in the Table LABEL:limits.

Table 1: Limits on the UHE photon flux used in the calculations, based on [10]. Energy [EeV] | Photon flux [km-2sr-1yr-1]
---|---
1 | 1.07
3 | 1.01$\times 10^{-2}$
10 | 4.15$\times 10^{-3}$
30 | 1.41$\times 10^{-3}$
100 | 1.05$\times 10^{-3}$

The final histogram of the signatures’ orientation for the energy spectra
1-100 EeV is shown in the Figure 4. It is normalized in such a way that the
number of all secondary photons cascades is 10 000. The uncertainties were
calculated from the Poisson distribution. For the dipole model, we cannot
include the UHE photons with energy 1 EeV due to the lack of data. Thus, the
anisotropy is more significant for the DQCS model. This result is very
promising, although it is visible only for a large amount of data. And on the
basis of used limits on the UHE photons flux, we should observed on the
observation sphere of the radius 10 000 km approximately 3000 signatures per
year, according to the dipole model of the solar magnetic field, or 17 000
signatures per year, according to the DQCS model. However, these numbers may
be underestimated, since we considered only UHE photons landing at the
observation sphere.

Figure 4: The distribution of signatures’ rotation for the whole energy
spectra 1-100 EeV of the UHE photons.

##  4 Conclusions

The obtained results indicate that the anisotropy of the spatial signatures
orientation may be used as an observable when the DQCS model is introduced in
order to simulate the solar magnetic field, which corresponds to UHE photons
approaching the Sun very close. This could help to decide whether the observed
photon emission from the vicinity of the Sun is related to the interaction of
the UHE photons with the solar magnetic field. However, for the dipole model,
a more detailed analysis and data including the 1 EeV UHE photons is required,
since the corresponding photons have the highest impact on the presence of the
signatures’ distribution anisotropies. Moreover, it necessary to investigate
the neglected cases in the simulation and precisely assess the fraction of
cascades giving observable CRE effects in the atmosphere, so further
optimisations of the simulator are required.

##  5 Acknowledgements

D.E.A.C. acknowledges support from the NCN OPUS Project No.
2018/29/B/ST2/02576. P.H. is supported by IRAP AstroCeNT (MAB/2018/7) funded
by FNP from ERDF. The CREDO Project is generously supported by the Visegrad
Fund under grant number 22220116.

## References

  * [1] A. A. Abdo et al. [Fermi-LAT], _Fermi-LAT Observations of Two Gamma-Ray Emission Components from the Quiescent Sun,_ _Astrophys. J._ 734, 116 (2011) [arXiv:1104.2093 [astro-ph.HE]]
  * [2] T. Linden, B. Zhou, J. F. Beacom, A. H. G. Peter, K. C. Y. Ng and Q. W. Tang, _Evidence for a New Component of High-Energy Solar Gamma-Ray Production,_ _Phys. Rev. Lett._ 121, no.13, 131103 (2018) [arXiv:1803.05436 [astro-ph.HE]]
  * [3] A. Albert et al. [HAWC], _First HAWC Observations of the Sun Constrain Steady TeV Gamma-Ray Emission,_ _Phys. Rev. D_ 98, no.12, 123011 (2018) [arXiv:1808.05620 [astro-ph.HE]]
  * [4] T. Erber, _High-energy electromagnetic conversion processes in intense magnetic fields_ , _Rev. Mod. Phys._ 38 626 (1966)
  * [5] J. K. Daugherty and A. K. Harding, _Pair production in superstrong magnetic fields_ , _Astrophys. J._ 273 761 (1983)
  * [6] N. Dhital, P. Homola, D. Gora, H. Wilczyński, K. A. Cheminant, G. Bhatta, T. Bretz, D. A. Castillo, A. Ćwikła and A. R. Duffy, et al., _Cosmic ray ensembles as signatures of ultra-high energy photons interacting with the solar magnetic field,_ _JCAP_ 03, no.03, 038 (2022) [arXiv:1811.10334 [astro-ph.HE]]
  * [7] P. Homola et al. [CREDO],, _Cosmic Ray Extremely Distributed Observatory_ , _Symmetry_ 12, no.11, 1835 (2020) [arXiv:2010.08351 [astro-ph.HE]]
  * [8] R. J. Leamon, S. W. McIntosh and A. M. Title _Deciphering Solar Magnetic Activity: The Solar Cycle Clock_ , _Front. Astron. Space Sci._ 9, 886670 (2022) [arXiv:2012.15186 [astro-ph.SR]]
  * [9] B. Bartoli et al. [ARGO-YBJ], _Search for gamma-ray emission from the Sun during solar minimum with the ARGO-YBJ experiment,_ _Astrophys. J._ 872, no.2, 143 (2019) [arXiv:1901.04201 [astro-ph.HE]]
  * [10] J. Rautenberg [Pierre Auger], _Limits on ultra-high energy photons with the Pierre Auger Observatory,_ _PoS ICRC2019_ 398 (2021)
  * [11] B. Poncyljusz, T. Bulik, N. Dhital, O. Sushchov, S. Stuglik, P. Homola, D. Alvarez-Castillo, M. Piekarczyk, T. Wibig and J. Stasielak, et al. _Simulation of the Isotropic Ultra-High Energy Photon Flux in the Solar Magnetic Field,_ _Universe_ 8, no.10, 498 (2022) [arXiv:2205.14266 [astro-ph.HE]]
  * [12] M. Banaszkiewicz, W. I. Axford and J. F. McKenzie, _An analytic solar magnetic field model_ , _Astronomy & Astrophysics_ 337 940 (1998)
  * [13] P. Homola, D. Góra, D. Heck, H. Klages, J. Pękala, M. Risse et al., _Simulation of ultra-high energy photon propagation in the geomagnetic field_ , _Comput. Phys. Commun._ 173 (2005) 71 [astro-ph/0311442]

## Full Authors List: CREDO Collaboration

David E. Alvarez Castillo1,4, Piotr Homola1,26, Oleksandr Sushchov1, Jarosław
Stasielak1, Sławomir Stuglik1, Dariusz Góra1, Vahab Nazari1, Cristina Oancea2,
Dmitriy Beznosko3, Noemi Zabari4, Alok C. Gupta5, Bohdan Hnatyk6, Alona
Mozgova6, Marcin Kasztelan7, Marcin Bielewicz7, Peter Kovacs8, Bartosz
Łozowski9, Mikhail V. Medvedev10,11, Justyna Miszczyk1, Łukasz Bibrzycki14,
Michał Niedźwiecki12, Katarzyna Smelcerz12, Tomasz Hachaj14, Marcin
Piekarczyk14, Maciej Pawlik13,14, Krzysztof Rzecki14, Matías Rosas15, Karel
Smolek16, Manana Svanidze17, Revaz Beradze17, Arman Tursunov18, Tadeusz
Wibig19, Jilberto Zamora-Saa20, Bożena Poncyljusz21, Justyna Mędrala22,
Gabriela Opiła22, Jerzy Pryga23, Ophir Ruimi24, Mario Rodriguez Cahuantzi25,
Niraj Dhital27.

1Institute of Nuclear Physics Polish Academy of Sciences, Radzikowskiego 152,
31-342 Kraków, Poland.
2ADVACAM, 12, 17000 Prague, Czech Republic
3Clayton State University, Morrow, Georgia, USA.
4Astroteq.ai Ltd., Juliusza Słowackiego 24, 35-069, Rzeszóow, Poland.
5Aryabhatta Research Institue of Observational Sciences (ARIES), Manora Peak,
Nainital 263001, India.
6Astronomical Observatory of Taras Shevchenko National University of Kyiv,
04053 Kyiv, Ukraine.
7National Centre for Nuclear Research, Andrzeja Soltana 7, 05-400 Otwock-
Świerk, Poland.
8Institute for Particle and Nuclear Physics, Wigner Research Centre for
Physics, 1121 Budapest, Konkoly-Thege Miklós út 29-33, Hungary.
9Faculty of Natural Sciences, University of Silesia in Katowice, Bankowa 9,
40-007 Katowice, Poland.
10Department of Physics and Astronomy, University of Kansas, Lawrence, KS
66045, USA.
11Laboratory for Nuclear Science, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA.
12Department of Computer Science, Cracow University of Technology, Warszawska
24, 31-155 Kraków, Poland.
13ACC Cyfronet AGH-UST, 30-950 Kraków, Poland.
14AGH University of Science and Technology, Mickiewicz Ave., 30-059 Kraków,
Poland.
15Liceo 6 Francisco Bauzá, Montevideo, Uruguay.
16Institute of Experimental and Applied Physics, Czech Technical University in
Prague.
17E. Andronikashvili Institute of Physics under Tbilisi State University,
Georgia.
18Institute of Physics, Silesian University in Opava, Bezručovo nám. 13,
CZ-74601 Opava, Czech Republic.
19University of Łódź, Faculty of Physics and Applied Informatics, 90-236 Łódź,
Pomorska 149/153, Poland.
20Universidad Andres Bello, Departamento de Ciencias Fisicas, Facultad de
Ciencias Exactas, Avenida Republica 498, Santiago, Chile.
21Faculty of Physics, University of Warsaw, 02-093 Warsaw, Poland.
22Faculty of Physics and Applied Computer Science, AGH University of Kraków,
Poland.
23 Pedagogical University of Krakow, Institute of Computer Science, ul.
Podchorążych, 30-084 Kraków, Poland.
24Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem, IL,
91904, Israel
25Facultad de Ciencias Físico Matemáticas-Benemérita Universidad Autónoma de
Puebla, 72570, Mexico
26AstroCeNT, Nicolaus Copernicus Astronomical Center Polish Academy of
Sciences, ul. Rektorska 4, 00-614 Warsaw, Poland. 27Central Department of
Physics, Tribhuvan University, Kirtipur 44613, Nepal