Lemma 8.2 of “Solvability of
Groups of Odd Order” by W. Feit and J. G. Thompson asserts that every finite
p-group P possesses a characteristic subgroup C of class at most 2 with the following
properties: (i) C∕Z(C) is elementary abelian, (ii) CP(C) = Z(C), and (iii)
[P,C] ⊆ Z(C). Subgroups of essentially the same type were used by Thompson in an
earlier paper “Normal p-complements for Finite Groups”. We shall call a subgroup
with these properties a critical subgroup of P.
If C is an arbitrary characteristic subgroup of P such that CP(C) = Z(C), it is easily
seen that any nontrivial p′-automorphism of P remains nontrivial when restricted to
C. This property of critical subgroups together with the restriction on their class are
the crucial ones for the applications. However, in the present note we shall show that
they can also be used to obtain a rather direct proof of a frequently quoted,
unpublished,1
theorem of Philip Hall which gives the structure of all p-groups having no noncyclic
characteristic abelian subgroups.
