# Abelian not implies contained in abelian subgroup of maximum order

From Groupprops

## Statement

It is possible to have a group of prime power order with an abelian subgroup such that is *not* contained in any Abelian subgroup of maximum order (?) in .

## Proof

### Example of a dihedral group

`Further information: dihedral group:D16`

Consider the dihedral group of order , specifically:

.

This has an Abelian subgroup of order eight: the cyclic subgroup generated by . Consider now the cyclic subgroup given as:

.

is a subgroup of order four.

Clearly, is self-centralizing in : . Thus, is not contained in a bigger Abelian subgroup, and hence is not contained in an Abelian subgroup of maximum order.