# Abelian not implies contained in abelian subgroup of maximum order

## Statement

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

## Proof

### Example of a dihedral group

Further information: dihedral group:D16

Consider the dihedral group of order $16$, specifically:

$P = \langle a,x \mid a^8 = x^2 = e, xax^{-1} = a^{-1} \rangle$.

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

$B := \langle a^4, x \rangle$.

$B$ is a subgroup of order four.

Clearly, $B$ is self-centralizing in $P$: $C_P(B) = B$. Thus, $B$ is not contained in a bigger Abelian subgroup, and hence $B$ is not contained in an Abelian subgroup of maximum order.