Every group is a union of maximal among abelian subgroups

Statement

Any group can be expressed as a union of subgroups, each of which is Maximal among abelian subgroups (?).

Definitions used

Maximal among Abelian subgroups

Further information: Maximal among Abelian subgroups

A maximal among abelian subgroups is an abelian subgroup that is not contained in any bigger abelian subgroup. Equivalently it is a subgroup that equals its own centralizer.

Related facts

• Every group is a union of cyclic subgroups

Facts used

1. Every abelian subgroup is contained in a maximal among abelian subgroups

Proof

Given: A slender group $G$.

To prove: $G$ is a union of subgroups, each of which is maximal among abelian subgroups.

Proof: We shall prove that every element $g\in G$ is contained in a subgroup of $G$ that is maximal among abelian subgroups of $G$. Taking the union of such subgroups over all $g\in G$ yields that $G$ is a union of subgroups, each of which is maximal among abelian subgroups.

First, observe that $g$ is contained in an abelian subgroup -- the cyclic subgroup generated by $g$. By fact (1), this Abelian subgroup is contained in a subgroup that is maximal among abelian subgroups. Hence $g$ is contained in a subgroup that is maximal among abelian subgroups of $G$.