Maximal orders of Abelian subgroups in finite simple groups.(English. Russian original)Zbl 0936.20009

Algebra Logika 38, No. 2, 131-160 (1999); translation in Algebra Logic 38, No. 2, 67-83 (1999).
Using the classification of the finite simple groups, the author proves that, for every Abelian subgroup $$A$$ of a finite simple non-Abelian group $$G$$ other than $$L_2(q)$$, the inequality $$|A|^3<|G|$$ holds. As a corollary, the following result is obtained: For a finite non-Abelian simple group $$G$$, there exist Abelian subgroups $$A,B$$ with $$G=ABA$$ if and only if $$G$$ is isomorphic to $$L_2(2^t)$$, $$t\geq 2$$. The latter result was announced earlier by D. L. Zagorin and L. S. Kazarin [Dokl. Math. 53, No. 2, 237-239 (1996); translation from Dokl. Akad. Nauk 347, No. 5, 590-592 (1996; Zbl 0895.20020)].

MSC:

 20D06 Simple groups: alternating groups and groups of Lie type 20D40 Products of subgroups of abstract finite groups 20D25 Special subgroups (Frattini, Fitting, etc.) 20D60 Arithmetic and combinatorial problems involving abstract finite groups

Zbl 0895.20020
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