Vol. 45, No. 1, 1973

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Local control and factorization of the focal subgroup

David Finkel

Vol. 45 (1973), No. 1, 113–128

There are a number of results that yield information about the structure of a finite group G, based on information about its local subgroups; that is, the normalizers of its p-subgroups. Alperin and Gorenstein defined a notion of control of transfer, fusion, or strong fusion in a group G by a conjugacy functor W, or rather by N(W(P)), where P is a Sylow p-subgroup of G, and proved that if W controls transfer, fusion or strong fusion in every local subgroup of G, then W controls, respectively, transfer, fusion or strong fusion in the entire group G.

This paper gives a definition of control by two functors, W and V , or rather N(W(P)) and C(V (P)), and proves that when W contains the center, and when V is contained in the center and satisfies another condition, control in certain local subgroups implies control in the entire group G. In particular, the result on transfer says that P G, the focal subgroup of G, can be factorized whenever their focal subgroups can be properly factorized. From theorems of Glauberman, Goldschmidt, and Thompson which give conditions under which a group can be factorized, it is possible to obtain results which say that the focal subgroup of G can be factorized whenever certain local subgroups are p-solvable and satisfy the hypotheses of the factorization theorems.

Mathematical Subject Classification 2000
Primary: 20D25
Received: 17 November 1971
Revised: 17 July 1972
Published: 1 March 1973
David Finkel