University of Isfahan Mathematics and Society 2345-6493 11 3 2026 01 11 Recognition of the group $E_{6}(5)$ by prime graph Recognition of the group $E_{6}(5)$ by prime graph 1 8 30163 10.22108/msci.2025.144850.1736 FA Zahra Momen Department of Mathematics Education, Farhangian University , P.O.Box 14665-889, Tehran, Iran Journal Article 2025 04 07 If \( n \) is an integer, the set of all prime divisors of \( n \) is denoted by \( \pi(n) \). Let \( G \) be a finite group. Then \( \pi(|G|) \) is denoted by \( \pi(G) \). The prime graph of \( G \), denoted by \( \Gamma(G) \), is constructed as follows: the vertex set is \( \pi(G) \), and two distinct primes \( p \) and \( q \) are connected by an edge if and only if \( G \) has an element of order \( pq \). A finite nonabelian simple group $P$ is called quasirecognizable by prime graph, if each finite group $G$ with $\Gamma(G) = \Gamma(P)$ has a unique composition factor isomorphic to $P$. We denote by $k(\Gamma(G))$ the number of isomorphism classes of finite groups $H$ satisfying $\Gamma(G)=\Gamma(H)$. Given a natural number $r$, a finite group $G$ is called $r$-recognizable by prime graph if $k(\Gamma(G))=r$ and if $k(\Gamma(G))=1$, is called recognizable . In this paper, as the main result, we show that if \( G \) is a finite group such that \( \Gamma(G) = \Gamma(E_6(5)) \), then \( G \cong E_6(5) \). If \( n \) is an integer, the set of all prime divisors of \( n \) is denoted by \( \pi(n) \). Let \( G \) be a finite group. Then \( \pi(|G|) \) is denoted by \( \pi(G) \). The prime graph of \( G \), denoted by \( \Gamma(G) \), is constructed as follows: the vertex set is \( \pi(G) \), and two distinct primes \( p \) and \( q \) are connected by an edge if and only if \( G \) has an element of order \( pq \). A finite nonabelian simple group $P$ is called quasirecognizable by prime graph, if each finite group $G$ with $\Gamma(G) = \Gamma(P)$ has a unique composition factor isomorphic to $P$. We denote by $k(\Gamma(G))$ the number of isomorphism classes of finite groups $H$ satisfying $\Gamma(G)=\Gamma(H)$. Given a natural number $r$, a finite group $G$ is called $r$-recognizable by prime graph if $k(\Gamma(G))=r$ and if $k(\Gamma(G))=1$, is called recognizable . In this paper, as the main result, we show that if \( G \) is a finite group such that \( \Gamma(G) = \Gamma(E_6(5)) \), then \( G \cong E_6(5) \). https://math-sci.ui.ac.ir/article_30163_5416e125b3ad24cb4329f39a658cbe3e.pdf