Introduction. Dedekind has determined all groups whose subgroups are all normal (see, e.g., [5, Theorem 12.5.4]). Partially generalizing this, Wielandt showed that a finite group is nilpotent, if and only if all its subgroups are subnormal, and also if and only if all maximal subgroups are normal [5, Corollary 10.3.1, 10.3.4]. Huppert [7, Sätze 23, 24] has shown that if all 2nd-maximal subgroup...

In 1] M. Aschbacher described the structure of the possible maximal subgroups of a classical group G with natural module M. Each such subgroup is either a member of a canonical class of subgroups, or it is the normalizer of a quasi-simple, absolutely irreducible subgroup H of G. In the latter case suppose that H is a classical group whose deening characteristic is coprime to that of G. The aim ...

Let $G$ be an infinite-dimensional representation-theoretic Kac--Moody group associated to a nonsingular symmetrizable generalized Cartan matrix. We consider Eisenstein series on induced from unramified cusp forms finite-dimensional Levi subgroups of maximal parabolic subgroups. Under natural condition subgroups, we prove that the cuspidal are entire full complex plane.

A subgroup $H$ is said to be $nc$-supplemented in a group $G$ if there exists a subgroup $Kleq G$ such that $HKlhd G$ and $Hcap K$ is contained in $H_{G}$, the core of $H$ in $G$. We characterize the supersolubility of finite groups $G$ with that every maximal subgroup of the Sylow subgroups is $nc$-supplemented in $G$.

Abstract Let G be a finite group and assume that of automorphisms A acts on , such the orders are relatively prime. We prove fact imposing certain conditions set maximal -invariant subgroups relating to nilpotency, p -nilpotency, normality or having p’ -order, determines properties structure as solubility, -solubility -nilpotency.

We call a group $G$ belongs to the class of groups $S_{p}'$, if for every $pd$-chief factor $A/B$ $G$, $((A/B)_{p})'=1$. In this paper, we investigate influence some second maximal subgroups which are related non-$c_{p}$-normal on structure $S_{p}'$.