Every group is a union of maximal among abelian subgroups

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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

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.