Maximal abelian normal subgroups in Nilpotent groups of class 2

TL;DR

This paper explores the structure of maximal abelian normal subgroups in nilpotent groups of class 2, revealing conditions under which these subgroups influence the overall group structure, especially with varying numbers of generators.

Contribution

The paper characterizes the structure of maximal abelian normal subgroups in finitely generated nilpotent groups of class 2, highlighting differences based on the number of generators.

Findings

In free nilpotent groups of class 2, such subgroups are cyclic extensions of the center.

Counterexamples exist for three-generator groups where the subgroup is not a cyclic extension.

Certain structural conditions are necessary for groups with four or more generators.

Abstract

A maximal abelian normal subgroup A in a nilpotent group N is self-centralizing. This makes their role an important one in determining the structure of the nilpotent group. For example if A is finite then N is also finite. In the free nilpotent group of class 2 such a group is always a cyclic extension of the center. However this need not be true in an arbitrary finitely generated class two nilpotent group. We show that three generator nilpotent groups violating this condition can be easily found. However in the case when there are four or more generators certain conditions must hold.