Abelian not implies contained in abelian subgroup of maximum order

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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.


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.