Abstract. A subgroup H of a group G is said to be SS-embedded in G if there exists a normal subgroup T of G such that HT is subnormal in G and H T H sG , where H sG is the maximal s- permutable subgroup of G contained in H. We say that a subgroup H is an SS-normal subgroup in G if there exists a normal subgroup T of G such that G = HT and H T H SS , where H SS is an SS-embedded subgroup of ...

the theorem 12 in [a note on $p$-nilpotence and solvability of finite groups, j. algebra 321(2009) 1555--1560.] investigated the non-abelian simple groups in which some maximal subgroups have primes indice. in this note we show that this result can be applied to prove that the finite groups in which every non-nilpotent maximal subgroup has prime index aresolvable.

Let G be a finitely generated group, and let mn(G) denote the number of maximal subgroups index n. An upper bound is given for degree subgroup growth all polycyclic metabelian groups (i.e., limsup log n, polynomial mn(G)). A condition when this attained. G=Zk⋊AZ, where generator Z on right acts (by conjugation) Zk by multiplication A∈GL(k,Z). It shown that grows like equal to blocks in rational...

suppose $g$ is a split connected‎ ‎reductive orthogonal or symplectic group over an infinite field‎ ‎$f,$ $p=mn$ is a maximal parabolic subgroup of $g,$ $frak{n}$ is‎ ‎the lie algebra of the unipotent radical $n.$ under the adjoint‎ ‎action of its stabilizer in $m,$ every maximal prehomogeneous‎ ‎subspaces of $frak{n}$ is determined‎.

The co-maximal subgroup graph $$\Gamma (G)$$ of a group G is whose vertices are non-trivial proper subgroups and two H K adjacent if $$HK=G$$ . In this paper, we continue the study , especially when has isolated vertices. We define new ^*(G)$$ which obtained by removing from characterize connected, complete graph, star an universal vertex etc. also find various parameters like diameter, girth, ...