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Finite Groups the Centralizers of whose involutions have Normal 2-Complements

Published online by Cambridge University Press:  20 November 2018

Daniel Gorenstein
Daniel Gorenstein*
Affiliation:
Northeastern University, Boston, Massachusetts
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In this paper we shall classify all finite groups in which the centralizer of every involution has a normal 2-complement. For brevity, we call such a group an I-group. To state our classification theorem precisely, we need a preliminary definition.

As is well-known, the automorphism group G = PΓL(2, q) of H= PSL(2, q), q= pn, is of the form G = LF, where L = PGL(2, q), L ⨞ G, F is cyclic of order n, L ∩ F = 1, and the elements of F are induced from semilinear transformations of the natural vector space on which GL(2, q) acts; cf. (3, Lemma 2.1) or (7, Lemma 3.3). It follows at once (4, Lemma 2.1; 8, Lemma 3.1) that the groups H and L are each I-groups. Moreover, when q is an odd square, there is another subgroup of G in addition to L that contains H as a subgroup of index 2 and which is an I-group.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 335 - 357
Copyright
Copyright © Canadian Mathematical Society 1969

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