Normal subgroup growth in free class-2-nilpotent groups

Normal subgroup growth in free class-2-nilpotent groups

Let F2,d denote the free class-2-nilpotent group on d generators. We compute the normal zeta functions ζ F2,d(s), prove that they satisfy local functional equations and determine their abscissae of convergence and pole orders.

Enumerating Finite Class-2-nilpotent Groups on 2 Generators

We compute the numbers g(n, 2, 2) of nilpotent groups of order n, of class at most 2 generated by at most 2 generators, by giving an explicit formula for the Dirichlet generating function P

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

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.  

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...