The Answers to a Problem and Two Conjectures about OD-Characterization of Finite Groups
{Ali Mahmoudifar \& Behrooz Khosravi
In [Akbari and Moghaddamfar, Recognizing by order and degree pattern of some projective special linear groups, {Internat. J. Algebra Comput., 2012] the authors possed the following problem: \\ {Problem. {Is there a simple group which is $k$-fold OD-characterizable for $k\geq3\ ?$ In this paper as the main result we give positive answer to the above problem and we introduce two simple groups which are $k$-fold OD-characterizable such that $k\geq6$. Also in [R. Kogani-Moghadam and A. R. Moghaddamfar, Groups with the same order and degree pattern, {Science China Mathematics, 2012], the authors possed two conjectures as follows: \\ {Conjecture 1. {All alternating groups $A_m$ with $m \not= 10$ are OD-characterizable. \\ {Conjecture 2. {All symmetric groups $S_m$, with $m \not= 10$, are $n$-fold OD-characterizable, where $n\in\{1, 3\$. In this paper we find some alternating and some symmetric groups such that these conjectures are not true for them.
1409.7903.tex
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\title{\bf The Answers to a Problem and Two Conjectures about OD-Characterization of Finite Groups }
\smallskip
\normalsize
\author{{\bf Ali Mahmoudifar  \&  Behrooz Khosravi}
\\ Dept. of Pure  Math.,  Faculty  of Math. and Computer Sci.,\\
Amirkabir University of Technology (Tehran Polytechnic),\\
424, Hafez Ave., Tehran 15914, IRAN, \\  e-mail:
khosravibbb@yahoo.com\\alimahmoudifar@gmail.com}
\begin{document}
\maketitle
\begin{abstract}
In [Akbari and Moghaddamfar, Recognizing by order and degree pattern of some projective special linear groups, {\it Internat. J. Algebra Comput.}, 2012]
the authors possed the following problem:
\\
{\bf Problem.} {\it Is there a simple group which is $k$-fold OD-characterizable for $k\geq3\ ?$ }

In this paper as the main result we give positive answer to the above problem and we introduce
two simple groups which are $k$-fold OD-characterizable such that $k\geq6$.

Also in [R. Kogani-Moghadam and A. R. Moghaddamfar, Groups with the same order and degree pattern,
{\it Science China Mathematics}, 2012], the authors possed two conjectures as follows:
\\
{\bf Conjecture 1.} {\it All alternating groups $A_m$ with $m \not= 10$ are OD-characterizable.}
\\
{\bf Conjecture 2.} {\it All symmetric groups $S_m$, with $m \not= 10$, are $n$-fold OD-characterizable, where
$n\in\{1, 3\}$.}

In this paper we find some alternating and some symmetric groups such that these conjectures are not true for them.
\end{abstract}
{\bf 2000 AMS Subject Classification}:  $20$D$05$, $20$D$60$,
20D08.
\\
{\bf Keywords :} Alternating group, prime graph, degree pattern, order, characterization.
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\section{Introduction}

In this paper every group is finite. If $n$ is a natural number, then we
denote by $\pi(n)$, the set of all prime divisors of $n$. If $G$ is a finite group, then $\pi(|G|)$ is denoted by $\pi(G)$.
The prime graph $GK(G)$ of a group $G$ is defined as a graph with vertex set $\pi(G)$ in
which two distinct primes $p,q\in\pi(G)$ are adjacent (and we write $p\sim q$) if G contains an element of order
$pq$. In {\rm\cite[Proposition~1.1]{anadja}}\label{adalt}, it is proved that if $r, s\in\pi(A_n)\setminus\{2\}$, then $r\nsim s$ in $GK(A_n)$ if and only if $r + s > n$ and
$2\nsim r$ in $GK(A_n)$ if and only if $r + 4 > n$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition}{\rm(\cite{mogh})}
Let $G$ be a group and $\pi(G)=\{p_1,p_2,\ldots,p_k\}$ where $p_1<p_2<\cdots<p_k$.
Then the degree pattern of $G$ is defined as follows:
$$D(G):=(\deg(p_1),\deg(p_2),\ldots,\deg(p_k)),$$
where $\deg(p_i)$, $1\leq i\leq k$, is the degree of vertex $p_i$ in $GK(G)$.
\end{definition}
For a group $G$, denote by $h_{OD}(G)$ the number of nonisomorphism finite groups $H$, such that $|H| = |G|$
and $D(H) = D(G)$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition}{\rm(\cite{mogh})}
Let $G$ be a group. We call that $G$ is $k$-fold OD-characterizable if $h_{OD}(G)=k$.
When $k=1$, the finite group $G$ is called OD-characterizable.
\end{definition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
OD-characterization of some finite groups are considered by some authors (see the references of \cite{mogh}). In \cite{hads}, the following problem
about OD-characterization of finite simple groups is possed (see also \cite{in,mogh,hads}):
 \begin{problem}\label{problem}
 Is there a simple group which is $k$-fold OD-characterizable for $k\geq3\ ?$
\end{problem}

Also in \cite{mogh}, the authors put the following conjectures:
\begin{conjecture}\label{conjecture1}
All alternating groups $A_m$ with $m\not= 10$ are OD-characterizable.
\end{conjecture}
\begin{conjecture}\label{conjecture2}
All symmetric groups $S_m$, with $m\not= 10$, are $n$-fold OD-characterizable, where
$n \in\{1,3\}$.
\end{conjecture}

Until now, Conjecture 1.4 is valid for $n=p$, $p+1$ and $p+2$,  where $p$ is an odd prime number.

In this paper as the main result we prove the following theorem and using this theorem we introduce some finite simple groups which are $k$-fold OD-characterizable, where $k\geq 6$. Also
we present some counterexamples for the above conjectures.
\begin{main}\label{main}
Let $p$ be an odd prime number such that $p+2$ and $p+4$ are not prime and $p+6=5^{\al}$, for some natural number $\al$. Also let
$H$ and $T$ be some groups such that $|H|=p+6=5^{\al}$ and $|T|=2(p+6)$. Then we have:
\begin{enumerate}
\item[{\rm (1)}] $|A_{p+6}|=|A_{p+5}\times H|$ and $D(A_{p+6})=D(A_{p+5}\times H)$,

\item[{\rm (2)}] $|S_{p+6}|=|S_{p+5}\times H|$ and $D(S_{p+6})=D(S_{p+5}\times H)$,

\item[{\rm (3)}]   $|S_{p+6}|=|A_{p+5}\times T|$ and $D(S_{p+6})=D(A_{p+5}\times T)$.
\end{enumerate}
\end{main}

\begin{remark}
Easily we can see that $p=619=5^4-6$, $15619=5^6-6$, $9765619=5^{10}-6$ satisfy the hypothesis of the main theorem.
\end{remark}


\begin{corollary}
If $(G,K)=(A_{625},A_{624})$ or $(S_{625},S_{624})$, then the following groups have the same order and degree pattern:
$G$, $K\times Z_{625}$, $K\times Z_{125}\times Z_{5}$, $K\times Z_{25}\times Z_{25}$, $K\times Z_{25}\times Z_5\times Z_5$, $K\times Z_5 \times Z_5 \times Z_5 \times Z_5$.
\end{corollary}
\begin{corollary}
We have $h_{OD}(A_{625})\geq6$. In Particular, the simple group $A_{625}$ is an answer for Problem{\rm~\ref{problem}} and also we see that Conjecture{\rm~\ref{conjecture1}} is not true.
\end{corollary}
%\begin{proof}
%By Corollary~1.7, it is obvious.
%\end{proof}
\begin{corollary}
We have $h_{OD}(S_{625})\geq6$. In Particular, Conjecture{\rm~\ref{conjecture2}} is not true.
\end{corollary}
\begin{corollary}
Let $p=5^{\al}-6$ satisfies the assumptions of the main theorem. If $m$ is the number of nonisomorphic finite groups of order
$5^{\al}$, then $h_{OD}(A_{p+6})\geq m+1$ and $h_{OD}(S_{p+6})\geq m+1$.
\end{corollary}
%\begin{proof}
%By Corollary~1.7, it is obvious.
%\end{proof}
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%\section{Preliminary Results}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{lemma}{\rm(\cite[Proposition~1.1]{anadja})}\label{adalt}
%Let $G = A_n$ be an alternating group of degree $n$.
%(1) Let $r, s\in\pi(G)$ be odd primes. Then $r$, $s$ are nonadjacent if and only if $r + s > n$.
%(2) Let $r \in\pi(G)$ be an odd prime. Then $2$, $r$ are nonadjacent if and only if $r + 4 > n$.
%\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{lemma}{\rm(\cite[Lemma~2.15]{mogh})}\label{moghadam}
%Let $S$ be a symmetric (resp., an alternating) group and let $G$ be a finite group with
%$\pi(G) = \pi(S)$ and $D(G) = D(S)$. Then the prime graph $GK(G)$ coincides with $GK(S)$.
%\end{lemma}
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\section{Proof of The Main Theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $p$ be a prime number such that $p+2$ and $p+4$ are not prime and $p+6=5^{\al}$, for some natural number $\al$.
The proof of the main theorem for symmetric groups is similar to alternating groups.

First we show that $GK(A_{p+6})=GK(A_{p+5})$. By assumption,  $p+6$ is a power of $5$ and so $\pi(A_{p+6})=\pi(A_{p+5})$. Since $p+2$ and $p+4$ are not prime numbers, by \cite[Proposition~1.1]{anadja}, we get that  every vertex is adjacent to $2$, $3$ and $5$ in $GK(A_{p+5})$ and $GK(A_{p+6})$. Then it is sufficient to check the adjacency of odd primes in these graphs.

Let $r,s\in \pi(A_{p+5})\setminus\{2\}$. By \cite[Proposition~1.1]{anadja}, if $r\sim s$ in $GK(A_{p+5})$, then $r+s\leq p+5<p+6$ and so $r\sim s$ in $GK(A_{p+6})$. Let $r\nsim s$, in $GK(A_{p+5})$. Then by \cite[Proposition~1.1]{anadja}, we have $r+s>p+5$. Since $p+6$ is odd, we get that $r+s\not=p+6$, and so $r+s>p+6$. This implies that $r\nsim s$ in $GK(A_{p+6})$.

Let $r,s\in\pi(A_{p+6})\setminus\{2\}$. If $r\nsim s$ in $GK(A_{p+6})$, then by \cite[Proposition~1.1]{anadja}, $r+s>p+6>p+5$ and so $r\nsim s$ in $GK(A_{p+5})$. Since $r+s\not=p+6$, if $r\sim s$ in $GK(A_{p+6})$, then $r+s\leq p+5$ and so $r\sim s$ in $GK(A_{p+5})$.

Therefore by the above discussion we conclude that $GK(A_{p+6})=GK(A_{p+5})$. Now if $H$ is a group such that $|H|=p+6=5^{\al}$, then $|A_{p+6}|=|A_{p+5}\times H|$. Also since $5$ is adjacent to each vertex in $GK(A_{p+6})$ and $|H|=5^{\al}$, then we get that $GK(A_{p+6})=GK(A_{p+5}\times H)$.
Therefore for every group $H$, such that $|H|=p+6=5^{\al}$, we have $|A_{p+6}|=|A_{p+5}\times H|$ and $D(A_{p+6})=D(A_{p+5}\times H)$.



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\begin{thebibliography}{99}

\bibitem{in}  B. Akbari, A. R. Moghaddamfar,  Recognizing by order and degree pattern of some projective special linear groups, {\it  Internat. J. Algebra Comput.}
 {\bf 22 (6)},  (2012)  1250051, 22 pp.


\bibitem{front} A. A. Hoseini, A. R. Moghaddamfar, Recognizing alternating groups $A_{p+3}$ for certain primes $p$ by their orders and degree patterns, {\it  Front Math. China,}
 {\bf 5 (3)}, (2010) 541--553.


\bibitem{mogh} R. Kogani-Moghadam and A. R. Moghaddamfar, Groups with the same order and degree pattern,
{\it Science China Mathematics, } {\bf 55 (4)}, (2012) 701--720.

\bibitem{hads} A. R. Moghaddamfar; A. R. Zokayi,  Recognizing finite groups through order and degree pattern, {\it Algebra Colloq.},{\bf  15 (3)}, (2008) 449-�456.

%
\bibitem{anadja} A. V. Vasil'ev and E. P. Vdovin, An adjacency criterion for the prime graph of a finite
simple group, {\it Algebra Logic, }{\bf 44 (6)}, (2005) 381--406.


\end{thebibliography}
\end{document}

1409.7903.extracted_bib
{in B. Akbari, A. R. Moghaddamfar, Recognizing by order and degree pattern of some projective special linear groups, {Internat. J. Algebra Comput. {22 (6), (2012) 1250051, 22 pp.
1409.7903.extracted_bib
{front A. A. Hoseini, A. R. Moghaddamfar, Recognizing alternating groups $A_{p+3$ for certain primes $p$ by their orders and degree patterns, {Front Math. China, {5 (3), (2010) 541--553.
1409.7903.extracted_bib
{mogh R. Kogani-Moghadam and A. R. Moghaddamfar, Groups with the same order and degree pattern, {Science China Mathematics, {55 (4), (2012) 701--720.
1409.7903.extracted_bib
{hads A. R. Moghaddamfar; A. R. Zokayi, Recognizing finite groups through order and degree pattern, {Algebra Colloq.,{15 (3), (2008) 449-�456. %
1409.7903.extracted_bib
{anadja A. V. Vasil'ev and E. P. Vdovin, An adjacency criterion for the prime graph of a finite simple group, {Algebra Logic, {44 (6), (2005) 381--406.