On the relation between the non-commuting graph and the prime graph

Document Type: Research Paper

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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.

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