Document Type : Research Paper

**Authors**

**Abstract**

$\pi(G)$ denote the set of prime divisors of the order of $G$ and denote by $Z(G)$ the center of $G$. The\textit{ prime graph} of $G$ is the graph with vertex set $\pi(G)$ where two distinct primes $p$ and $q$ are joined by an edge if and only if $G$ contains an element of order $pq$ and the \textit{non-commuting graph} of $G$ is the graph with the vertex set $G-Z(G)$ where two non-central elements $x$ and $y$ are joined by an edge if and only if $xy \neq yx$.

Let $ G $ and $ H $ be non-abelian finite groups with isomorphic non-commuting graphs. In this article, we show that if $ | Z ( G ) | = | Z ( H ) | $, then $ G $ and $ H $ have the same prime graphs and also, the set of orders of the maximal abelian subgroups of $ G $ and $ H $ are the same.

Let $ G $ and $ H $ be non-abelian finite groups with isomorphic non-commuting graphs. In this article, we show that if $ | Z ( G ) | = | Z ( H ) | $, then $ G $ and $ H $ have the same prime graphs and also, the set of orders of the maximal abelian subgroups of $ G $ and $ H $ are the same.

**Keywords**

**Main Subjects**

A. Abdollahi, S. Akbari and H.R. Maimani (2006). Non-commuting graph of a group. *J. Algebra*. 298 (2), 468-492 N. Ahanjideh and A. Iranmanesh On the orders of maximal abelian subgroups of
Bn(q). G.Y. Chen (2006). A characterization of alternating groups by the set of orders of their maximal abelian subgroups. *Siberian Math. J.*. 47 (3), 594-596 M.R. Darafsheh (2009). Groups with the same non-commuting graph. *Discrete Appl. Math.*. 157 (4), 833-837 A. Iranmanesh and A. Jafarzadeh (2008). On the commuting graph associated with the symmetric and alternating groups. *J. Algebra Appl.*. 7 (1), 129-146 E.I. Khukhro and V.D. Mazurov (Editors) (2010). Unsolved problems in group theory: The Kourovka Notebook. *Sobolev Institute of Mathematics, Novosibirsk*. 17th edition A.S. Kondratev (1990). Prime graph components of finite simple groups. *Math. USSR-Sb.*. 67 (1), 235-247 A.V. Vasil'ev (2005). On a relation between the structure of a
finite group and the properties of its prime graph. *Siberian Math. J.*. 46 (3), 396-404 A.V. Vasil'ev and E. P. Vdovin (2005). An adjacency criterion in the prime graph of a finite simple group. *Algebra Logic*. 44 (6), 381-406 J.S. Williams (1981). Prime graph components of finite groups. *J. Algebra*. 69 (2), 487-513

March 2012

Pages 25-28

**Receive Date:**09 July 2011**Revise Date:**07 November 2011**Accept Date:**07 November 2011**Published Online:**01 March 2012