# Every group is a union of maximal among abelian subgroups

## Contents

## 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

## Proof

**Given**: A slender group .

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

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

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