In this paper we find a necessary and sufficient condition for a finite nilpotent group to have an abelian central automorphism group.

‎let $g$ be a finite group and $z(g)$ be the center of $g$‎. ‎for a subset $a$ of $g$‎, ‎we define $k_g(a)$‎, ‎the number of conjugacy classes of $g$ which intersect $a$ non-trivially‎. ‎in this paper‎, ‎we verify the structure of all finite groups $g$ which satisfy the property $k_g(g-z(g))=5$ and classify them‎.

it is inferred from the symmetrical and luminous x-ray emission of fossil groups that they are mature, relaxed galaxy systems. cosmological simulations and observations focusing on their dark halo and inter-galactic medium properties confirm their early formation. recent photometric observations suggest that, unlike the majority of non-fossil brightest group galaxies (bggs), the central early-t...

a $p$-group $g$ is $p$-central if $g^{p}le z(g)$‎, ‎and $g$ is‎ ‎$p^{2}$-abelian if $(xy)^{p^{2}}=x^{p^{2}}y^{p^{2}}$ for all $x,yin‎ ‎g$‎. ‎we prove that for $g$ a finite $p^{2}$-abelian $p$-central ‎$p$-group‎, ‎excluding certain cases‎, ‎the order of $g$ divides the ‎order of $text{aut}(g)$‎.

a $p$-group $g$ is $p$-central if $g^{p}le z(g)$‎, ‎and $g$ is‎ ‎$p^{2}$-abelian if $(xy)^{p^{2}}=x^{p^{2}}y^{p^{2}}$ for all $x,yin‎ ‎g$‎. ‎we prove that for $g$ a finite $p^{2}$-abelian $p$-central‎ ‎$p$-group‎, ‎excluding certain cases‎, ‎the order of $g$ divides the‎ ‎order of $text{aut}(g)$‎.