Let G be a finite p-group and let Aut_l(G) be the group of absolute central automorphisms of G. In this paper we give necessary and sufficient condition on G such that each absolute central automorphism of G fixes the centre element-wise. Also we classify all groups of orders p^3 and p^4 whose absolute central automorphisms fix the centre element-wise.

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

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 $gcong l_2(81)$‎.