Vol. 19, No. 1, 1966

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On a theorem of Philip Hall

Daniel E. Gorenstein

Vol. 19 (1966), No. 1, 77–80

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in the present note we shall show that they can also be used to obtain a rather direct proof of a frequently quoted, unpublished,1 theorem of Philip Hall which gives the structure of all p-groups having no noncyclic characteristic abelian subgroups.

Mathematical Subject Classification
Primary: 20.40
Received: 8 December 1965
Published: 1 October 1966
Daniel E. Gorenstein