Let $G$ be a $p$-group of order $p^n$ and $Phi$=$Phi(G)$ be the Frattini subgroup of $G$. It is shown that the nilpotency class of $Autf(G)$, the group of all automorphisms of $G$ centralizing $G/ Fr(G)$, takes the maximum value $n-2$ if and only if $G$ is of maximal class. We also determine the nilpotency class of $Autf(G)$ when $G$ is a finite abelian $p$-group.

let $g$ be a group and $h$ a subgroup of $g$‎. ‎ $h$ is said to have semi-$pi$-property in $g$ if there is a subgroup $t$ of $g$ such that $g=ht$ and $hcap t$ has $pi$-property in $t$‎. ‎in this paper‎, ‎investigating on semi-$pi$-property of subgroups‎, ‎we shall obtain some new description of finite groups‎.

Abstract We provide the structure of unit group $${\mathbb {F}}_{p^k}S_n$$ F p k S n , where $$p>n$$ > is a prime and $$S_n$$ denotes symmetric on n l...