Groups that are pairwise nilpotent

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

Generalized Heineken–Mohamed type groups

We prove that a torsion group G with all subgroups subnormal is a nilpotent group or G = N(A1×· · ·×An) is a product of a normal nilpotent subgroup N and pi -subgroups Ai , where Ai = A (i) 1 · · ·A (i) mi G , A (i) j is a Heineken–Mohamed type group, and p1, . . . , pn are pairwise distinct primes (n ≥ 1; i = 1, . . . , n; j = 1, . . . ,mi and mi are positive integers).

Nilpotent groups with three conjugacy classes of non-normal subgroups

‎Let $G$ be a finite group and $nu(G)$ denote the number of conjugacy classes of non-normal subgroups of $G$‎. ‎In this paper‎, ‎all nilpotent groups $G$ with $nu(G)=3$ are classified‎.  

NILPOTENCY AND SOLUBILITY OF GROUPS RELATIVE TO AN AUTOMORPHISM

In this paper we introduce the concept of α-commutator which its definition is based on generalized conjugate classes. With this notion, α-nilpotent groups, α-solvable groups, nilpotency and solvability of groups related to the automorphism are defined. N(G) and S(G) are the set of all nilpotency classes and the set of all solvability classes for the group G with respect to different automorphi...

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 ...