let $pi_e(g)$ be the set of element orders of a finite group $g$‎. ‎let $nse(g)={m_nmid ninpi_e(g)}$‎, ‎where $m_n$ be the number of elements of order $n$ in $g$‎. ‎in this paper‎, ‎we prove that if $nse(g)=nse(l_2(81))$‎, ‎then $gcong l_2(81)$‎.

let $pi_e(g)$ be the set of element orders of a finite group $g$‎. ‎let $nse(g)={m_nmid ninpi_e(g)}$‎, ‎where $m_n$ be the number of elements of order $n$ in $g$‎. ‎in this paper‎, ‎we prove that if $nse(g)=nse(l_2(81))$‎, ‎then $gcong l_2(81)$‎.

in this paper we show that if q is a power of a prime p , then the projective special linear group psl(2, q) and the stabilizer of a point of the projective line have maximum sum element orders among all proper subgroups of projective general linear group pgl(2, q) for q odd and even respectively

For a finite group [Formula: see text], let text] denote the sum of element orders text]. If is positive integer be cyclic order It known that maximum value on set groups and if only In this paper, we investigate second largest structure with when odd.

Let $G$ be a finite group and $\psi(G) = \sum_{g \in G} o(g)$, where $o(g)$ denotes the order of $g G$. First, we prove that if is $n$ >31\psi(C_n)/77$, $C_n$ cyclic $n$, then supersolvable. This proves conjecture M.~{T\u{a}rn\u{a}uceanu}. Moreover, M. Herzog, P. Longobardi Maj put forward following conjecture: If $H\leq G$, \leqslant \psi(H) |G:H|^2$. In sequel, by an example show this not sat...

Let $ G be a periodic group, and let \omega(G)\subseteq{