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.

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**Main Subjects**

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