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Let $G$ be a finite group and ${\pi }_e\left(G\right)$ be the set of element orders of $G$. Let $k\in {\pi }_e\left(G\right)$ and $m_k$ be the number of elements of order $k$ in $G$. Set $\mathrm{nse}\left(G\right):=\{m_k:k\in {\pi }_e\left(G\right)\}$. In fact $\mathrm{nse}\left(G\right)$ is the set of sizes of elements with the same order in $G$. In this paper, by $\mathrm{nse}\left(G\right)$ and order, we give a new characterization of finite projective special linear groups $L_2\left(p\right)$ over a field with $p$ elements, where $p$ is prime. We prove the following theorem: If $G$ is a group such that $\left|G\right|=|L_2\left(p\right)|$ and $\mathrm{nse}\left(G\right)$ consists of $1$, $p^2-1$, $p(p+ϵ)/2$ and some numbers divisible by $2p$, where $p$ is a prime greater than...

Let $G$ be a finite group, and let $N\left(G\right)$ be the set of conjugacy class sizes of $G$. By Thompson’s conjecture, if $L$ is a finite non-abelian simple group, $G$ is a finite group with a trivial center, and $N\left(G\right)=N\left(L\right)$, then $L$ and $G$ are isomorphic. Recently, Chen et al. contributed interestingly to Thompson’s conjecture under a weak condition. They only used the group order and one or two special conjugacy class sizes of simple groups and characterized successfully sporadic simple groups (see Li’s PhD dissertation). In...

Let $G$ be a finite group. The main supergraph $𝒮\left(G\right)$ is a graph with vertex set $G$ in which two vertices $x$ and $y$ are adjacent if and only if $o\left(x\right)\mid o\left(y\right)$ or $o\left(y\right)\mid o\left(x\right)$. In this paper, we will show that $G\cong Sz\left(q\right)$ if and only if $𝒮\left(G\right)\cong 𝒮\left(Sz\right(q\left)\right)$, where $q=2^{2m+1}\ge 8$.

We prove that if the average number of Sylow subgroups of a finite group is less than $\frac{41}{5}$ and not equal to $\frac{29}{4}$, then $G$ is solvable or $G/F\left(G\right)\cong A_5$. In particular, if the average number of Sylow subgroups of a finite group is $\frac{29}{4}$, then $G/N\cong A_5$, where $N$ is the largest normal solvable subgroup of $G$. This generalizes an earlier result by Moretó et al.