In this paper, we prove that Ree group ${}^2G_2(q)$, where $qpmsqrt{3q}+1$ is a prime number can be uniquely determined by the order of group and the number of elements with the same order.

In this paper, we prove that Ree group ${}^2G_2(q)$, where $qpmsqrt{3q}+1$ is a prime number can be uniquely determined by the order of group and the number of elements with the same order.

The set of element orders of a finite group $G$ is called the {em spectrum}. Groups with coinciding spectra are said to be {em isospectral}. It is known that if $G$ has a nontrivial normal soluble subgroup then there exist infinitely many pairwise non-isomorphic groups isospectral to $G$. The situation is quite different if $G$ is a nonabelain simple group. Recently it was proved that if $L$ is a simple classical group of dimension at least 62 and $G$ is a finite group isospectral to $L$, then up to isomorphism $Lleq GleqAut L$. We show that the assertion remains true if 62 is replaced by 38.

For a finite group $G$, let $nu_{nc}(G)$ denote the number of conjugacy classes of non-normal non-cyclic subgroups of $G$. We characterize the finite non-nilpotent groups whose all non-normal non-cyclic subgroups are conjugate.

Let $G$ be a finite group. We consider the set of the irreducible complex characters of $G$, namely $Irr(G)$, and the related degree set $cd(G)={chi(1) : chiin Irr(G)}$. Let $rho(G)$ be the set of all primes which divide some character degree of $G$. In this paper we introduce the bipartite divisor graph for $cd(G)$ as an undirected bipartite graph with vertex set $rho(G)cup (cd(G)setminus{1})$, such that an element $p$ of $rho(G)$ is adjacent to an element $m$ of $cd(G)setminus{1}$ if and only if $p$ divides $m$. We denote this graph simply by $B(G)$. Then by means of combinatorial properties of this graph, we discuss the structure of the group $G$. In particular, we consider the cases where $B(G)$ is a path or a cycle.

$M_{24}$ is the largest Mathieu sporadic simple group of order $244 823 040 = 2^{10} {cdot} 3^3 {cdot} 5 {cdot} 7 {cdot} 11 {cdot} 23$ and contains all the other Mathieu sporadic simple groups as subgroups. The object in this paper is to study the ranks of $M_{24}$ with respect to the conjugacy classes of all its nonidentity elements.