let $g$ be a finite group‎. ‎a subgroup‎ ‎$h$ of $g$ is called an $mathcal{h}$-subgroup in‎ ‎$g$ if $n_g(h)cap h^{g}leq h$ for all $gin‎ ‎g$. a subgroup $h$ of $g$ is called a weakly‎ ‎$mathcal{h}^{ast}$-subgroup in $g$ if there exists a‎ ‎subgroup $k$ of $g$ such that $g=hk$ and $hcap‎ ‎k$ is an $mathcal{h}$-subgroup in $g$. we‎ ‎investigate the structure of the finite group $g$ under the‎ ‎as...

This paper proves: Let F be a saturated formation containing U . Suppose that G is a group with a normal subgroup H such that G/H ∈ F . (1) If all maximal subgroups of any Sylow subgroup of F ∗(H) are c-supplemented in G, then G ∈ F ; (2) If all minimal subgroups and all cyclic subgroups with order 4 of F ∗(H) are c-supplemented in G, then G ∈ F .

1 Multiply transitive groups Theorem 1.1. Let Ω be a finite set and G ≤ Sym(Ω) be 2–transitive. Let N E G be a minimal normal subgroup. Then one of the following holds: (a) N is regular and elementary abelian. (b) N is primitive, simple and not abelian. Proof. First we show that N is unique. Suppose that M is another minimal normal subgroup of G, so N ∩M = {e} and therefore [N,M ] = {e}. Since ...

The purpose of this paper is to present a method for translating the problem of finding all maximal subgroups of finite groups into questions concerning groups that are nearly simple. (A finite group is called nearly simple if it has only one minimal normal subgroup and that it nonabelian and simple.) In view of the recently announced classification of all finite simple groups this seems to be ...

The covering number of a finite group [Formula: see text], denoted is the smallest positive integer text] such that union proper subgroups. We calculate for family primitive groups with unique minimal normal subgroup isomorphic to divisible by and cyclic. This generalization result Swartz concerning symmetric groups. also prove an asymptotic pairwise generation.

Abstract A subgroup functor τ \tau is said Φ \Phi -regular if for all primitive groups G G , whenever H ∈ ( ) H\in \left(G) a p p -subgroup and N N minimal normal of then <m:mo...