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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.
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