Let $G$ be a finite group, let $p$ prime and ${\rm Pr}_p(G)$ the probability that two random $p$-elements of commute. In this paper we prove Pr}_p(G) > (p^2+p-1)/p^3$ if only has normal abelian Sylow $p$-subgroup, which generalizes previous results on widely studied commuting group. This bound is best possible in sense for each there are groups with = classify all such groups. Our proof based b...

we say that a finite group $g$ is conjugacy expansive if for anynormal subset $s$ and any conjugacy class $c$ of $g$ the normalset $sc$ consists of at least as many conjugacy classes of $g$ as$s$ does. halasi, mar'oti, sidki, bezerra have shown that a groupis conjugacy expansive if and only if it is a direct product ofconjugacy expansive simple or abelian groups.by considering a character analo...

let $g$ be a finite group‎. ‎we denote by $psi(g)$ the integer $sum_{gin g}o(g)$‎, ‎where $o(g)$ denotes the order of $g in g$‎. ‎here we show that‎ ‎$psi(a_5)< psi(g)$ for every non-simple group $g$ of order $60$‎, ‎where $a_5$ is the alternating group of degree $5$‎. ‎also we prove that $psi(psl(2,7))