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 fact about::maximal among abelian subgroups.

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) uses::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 $$G$$.