On a theorem of Philip Hall
Daniel Gorenstein · Pacific Journal of Mathematics · 1966
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) Cβ(C) is elementary abelian, (ii) MO = 3(C), and (iii) [P, C] e 3(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 &i>(C) = 3(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in the present note we shall show that they can also be used to obtain a rather direct proof of a frequently quoted, unpublished, 1 theorem of Philip Hall which gives the structure of all pgroups having no noncyclic characteristic abelian subgroups.