we call $h$ an $ss$-embedded subgroup of $g$ if there exists a‎ ‎normal subgroup $t$ of $g$ such that $ht$ is subnormal in $g$ and‎ ‎$hcap tleq h_{sg}$‎, ‎where $h_{sg}$ is the maximal $s$-permutable‎ ‎subgroup of $g$ contained in $h$‎. ‎in this paper‎, ‎we investigate the‎ ‎influence of some $ss$-embedded subgroups on the structure of a‎ ‎finite group $g$‎. ‎some new results were obtained.

let $g$ be a finite group‎. ‎a subset $x$ of $g$ is a set of pairwise non-commuting elements‎ ‎if any two distinct elements of $x$ do not commute‎. ‎in this paper‎ ‎we determine the maximum size of these subsets in any finite‎ ‎non-abelian metacyclic $2$-group and in any finite non-abelian $p$-group with an abelian maximal subgroup‎.

We give an explicit list of all p-groups G with a cyclic subgroup of index p2, such that the group algebra KG over the field K of characteristic p has a filtered multiplicative K-basis. We also proved that such a K-basis does not exist for the group algebra KG, in the case when G is either a powerful p-group or a two generated p-group ( p 6= 2 ) with a central cyclic commutator subgroup. This p...

Let G be a finite p-group and let d(G) denote the cardinality of a minimal generating set of G. G is said to be d-maximal if d(H) < d(G) for any proper subgroup H of G. In this paper we show that if G is a d-maximal p-group where p is odd, then G has exponent p or p2. For p = 2, we show that under certain conditions, a d-maximal 2-group has exponent four. We then give some classes of d-maximal ...