Use the preceding exercise to prove that if is a 2-group which has a cyclic center and is a subgroup of index 2 in , then has rank at most 2.
Note that is normal and that is characteristic; thus is normal in . Moreover, letting , we know that is characteristic, so that is normal. Moreover, is elementary abelian by construction. Note that, by §5.2 #7, and have the same rank.
If , then conjugation by is an automorphism of . If we choose , and let be fixed by – that is, , then we have (since ) and . Since , in fact . Thus the subgroup of which is fixed under conjugation by is contained in , and thus has rank 1.
By this previous exercise, has rank at most 2, so that also has rank at most 2.