Hostname: page-component-77c89778f8-gq7q9 Total loading time: 0 Render date: 2024-07-21T08:41:35.414Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

1. Alperin, J. L., Sylow intersection and fusion, J. Algebra 6 (1967), 222–241.Google Scholar
2. Glauberman, G., A characteristic subgroup of a p-stable group, Can. J. Math. 20 (1968), 1101–1135.Google Scholar
3. Glauberman, G., A characterization of the Suzuki groups (to appear).Google Scholar
4. Gorenstein, D., Finite groups in which Sylow 2-subgroups are abelian and centralizers of involutions are solvable, Can. J. Math. 17 (1965), 860–896.Google Scholar
5. Gorenstein, D., Finite groups, (New York, 1968).Google Scholar
6. Gorenstein, D. and Walter, J. H., On finite groups with dihedral Sylow 2-subgroups, Illinois J. Math. 6 (1962), 553–593.Google Scholar
7. Gorenstein, D. and Walter, J. H., Qn ihe maximal subgroups of finite simple groups, J. Algebra 1 (1964), 168–213.Google Scholar
8. Gorenstein, D. and Walter, J. H., The characterization of finite groups with dihedral Sylow 2-subgroups. I, II, III, J. Algebra 2 (1965), 85-151, 218-270, 334–393.Google Scholar
9. Higman, G., Odd characterizations of finite groups, mimeographed booklet, Oxford University, 1967.Google Scholar
10. Janko, Z. and Thompson, J. (to appear).Google Scholar
11. Suzuki, M., Finite groups with nilpotent centralizers, Trans. Amer. Math. Soc. 99 (1961), 425–470.Google Scholar
12. Suzuki, M., On a class of doubly transitive groups, Ann. of Math. (2) 75 (1962), 105–145.Google Scholar
13. Suzuki, M., Finite groups in which the centralizer of any element of order 2 is 2-closed, Ann. of Math. (2) 82 (1965), 191–212.Google Scholar
14. Zassenhaus, H., Kennzeichnung endlicher linearer Gruppen als Per mutations gruppen, Abh. Math. Sem. Univ. Hamburg 11 (1936), 17–40.Google Scholar