\titleofpresentation{Groups with Symmetric Non-commuting Graphs}

\presenter{Alireza  Abdollahi} 

Problem 31 of \cite{abdollahiPC} is the
following:
\begin{question}\label{abdollahicam}
Which finite groups have the property that the automorphism group
act transitively on the set of ordered pairs of non-commuting
elements?
\end{question}
   Every abelian group trivially
satisfies the property stated in Question \ref{abdollahicam}.
 Let $G$ be a finite non-abelian group satisfying the property stated in Question \ref{abdollahicam}. This property
means that the non-commuting graph $\Gamma_G$ is
$Aut(G)$-symmetric, where $Aut(G)$ is the automorphism group of
$G$, $\Gamma_G$ is the graph whose vertex set is $G\backslash
Z(G)$ ($Z(G)$ is the center of $G$) and the edge set is the set
of all non-commuting pairs of elements of $G$ (see \cite{abdollahiAAM} or
\cite{abdollahiMSZZ}) and recall that a graph $\Gamma$ is called
$K$-symmetric for a subgroup $K$ of $Aut(\Gamma)$ if $K$ acts
transitively on the set of ordered pairs of adjacent vertices of
$\Gamma$.

Pablo Spiga has pointed out that the group $G$  must be a
$p$-group of Frattini class 2, for some prime $p$ (see
\cite{abdollahiPC}). We generalize this result by proving that the group
$G$ is  nilpotent of class 2 and $G/Z(G)$ is  elementary abelian
(see Theorem \ref{abdollahisym}, below). Our main results are
\begin{theorem}\label{abdollahisym}
Let $G$ be a  finite non-abelian  group having the property that
the automorphism group act transitively on the set of ordered
pairs of non-commuting elements. Then
\begin{enumerate}
\item $Aut(G)$ acts transitively on $G\backslash Z(G)$.
\item for all $x,y\in G\backslash Z(G)$, we have $|x|=|y|$.
\item for all $x,y\in G\backslash Z(G)$, we have $C_G(x)\cong
C_G(y)$.
\item every two non-abelian 2-generator subgroups of $G$ are
isomorphic and every non-abelian 2-generator subgroup $H$ of $G$
has the same property like $G$, i.e.,  the automorphism group of
$H$ act transitively on the set of ordered pairs of non-commuting
elements of $H$.
\item $G$ is a $p$-group for some prime $p$.
\item $\Phi(G)\leq Z(G)$ ($\Phi(G)$ denotes the Frattini subgroup of $G$). In particular, $G$ is nilpotent of class
$2$ and $\frac{G}{Z(G)}$ is elementary abelian.
\end{enumerate}
\end{theorem}
\begin{theorem}\label{abdollahi2-gen}
A  finite non-abelian $2$-generator group $G$ has the property
that the automorphism group acts transitively on the set of
ordered pairs of non-commuting elements  if and only if $G$ is
either isomorphic to $$\mathcal{G}=\left<x,y \;|\;
x^{p^n}=y^{p^n}=[x,y]^p=[x,y,y]=[x,y,x]=1\right>,$$ for some
prime number  $p$  and integer $n>0$ or $G\cong Q_8$, the
quaternion group of order 8.
\end{theorem}
\begin{theorem}
Let $G$ be a non-abelian $3$-generator group having the property
that the automorphism group acts transitively on the set of
ordered pairs of non-commuting elements.  Then $C_G(x)=\langle x
\rangle Z(G)$ for all $x\in G\backslash Z(G)$.
\end{theorem}
\noindent {\bf Acknowledgement.} The author thanks the Center of
Excellence for Mathematics, University of Isfahan.
\begin{ourbiblio}{99}
\bibitem{abdollahiAAM} A. Abdollahi, A. Akbari and H.R. Maimani,
Non-commuting graph of a group, J. Algebra {\bf 298} (2006),
468-492.
\bibitem{abdollahiMSZZ} A. R. Moghaddamfar, W. Shi, W. Zhou, A. R. Zokayi,  On the noncommuting graph associated with a
finite group, Siberian Math. J. {\bf 46} No. 2, (2005) 325-332.
\bibitem{abdollahiPC} Peter Cameron's Home page, Old Problems,
\url{http://www.maths.qmul.ac.uk/~pjc/oldprob.html}.
\end{ourbiblio}
\presenteraddress{Department of Mathematics; University of
Isfahan; Isfahan 81746-73441; Iran}

\email{a.abdollahi@math.ui.ac.ir} 

\web{www.dr-abdollahi.ir}


