Geodesic
Finite Groups the Centralizers of whose involutions have Normal 2-Complements
Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 335-357

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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. 
Gorenstein, Daniel. Finite Groups the Centralizers of whose involutions have Normal 2-Complements. Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 335-357. doi: 10.4153/CJM-1969-035-x
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[1] 1. Alperin, J. L., Sylow intersection and fusion, J. Algebra 6 (1967), 222–241. Google Scholar

[2] 2. Glauberman, G., A characteristic subgroup of a p-stable group, Can. J. Math. 20 (1968), 1101–1135. Google Scholar

[3] 3. Glauberman, G., A characterization of the Suzuki groups (to appear). Google Scholar

[4] 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] 5. Gorenstein, D., Finite groups, (New York, 1968). Google Scholar

[6] 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] 7. Gorenstein, D. and Walter, J. H., Qn ihe maximal subgroups of finite simple groups, J. Algebra 1 (1964), 168–213. Google Scholar

[8] 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] 9. Higman, G., Odd characterizations of finite groups, mimeographed booklet, Oxford University, 1967. Google Scholar

[10] 10. Janko, Z. and Thompson, J. (to appear). Google Scholar

[11] 11. Suzuki, M., Finite groups with nilpotent centralizers, Trans. Amer. Math. Soc. 99 (1961), 425–470. Google Scholar

[12] 12. Suzuki, M., On a class of doubly transitive groups, Ann. of Math. (2) 75 (1962), 105–145. Google Scholar

[13] 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] 14. Zassenhaus, H., Kennzeichnung endlicher linearer Gruppen als Per mutations gruppen, Abh. Math. Sem. Univ. Hamburg 11 (1936), 17–40. Google Scholar

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