Finite groups with a nilpotent maximal subgroup

Finite Groups With a Certain Number of Elements Pairwise Generating a Non-Nilpotent Subgroup

finite groups with a certain number of elements pairwise generating a non-nilpotent subgroup

Finite Groups with a Certain Number of Elements Pairwise Generating a Non-nilpotent Subgroup

Let n > 0 be an integer and X be a class of groups. We say that a group G satisfies the condition (X , n) whenever in every subset with n + 1 elements of G there exist distinct elements x, y such that 〈x, y〉 is in X . Let N and A be the classes of nilpotent groups and abelian groups, respectively. Here we prove that: (1) If G is a finite semi-simple group satisfying the condition (N , n), then ...

FINITE GROUPS WITH A SYSTEM OF NILPOTENT SUBGROUPS CONTAINING THE SYLOW SUBGROUP By

Let G be a finite group and p an odd prime. By M < G we denote that M is a proper subgroup of G. Put the set Ψ p (G) = {M:M < G, |G : M| 6= a prime power and |G : M|p = 1}. In this paper we investigate the structure of G if every member of Ψ p (G) is nilpotent.

Relative non-Normal Graphs of a Subgroup of Finite Groups

Let G be a finite group and H,K be two subgroups of G. We introduce the relative non-normal graph of K with respect to H , denoted by NH,K, which is a bipartite graph with vertex sets HHK and KNK(H) and two vertices x ∈ H HK and y ∈ K NK(H) are adjacent if xy / ∈ H, where HK =Tk∈K Hk and NK(H) = {k ∈ K : Hk = H}. We determined some numerical invariants and state that when this graph is planar or...