Large subgroups of simple groups. (English) Zbl 1308.20012
The problem of determining the “large” maximal subgroups of finite simple groups has a long history (see, for example, M. W. Liebeck [Proc. Lond. Math. Soc. (3) 50, 426-446 (1985; Zbl 0591.20021); ibid. 55, 299-330 (1987; Zbl 0627.20026)], A. Maróti [J. Algebra 258, No. 2, 631-640 (2002; Zbl 1018.20002)]), with many applications (see, for example, [M. W. Liebeck and J. Saxl, Bull. Lond. Math. Soc. 18, 165-172 (1986; Zbl 0586.20003)], [M. W. Liebeck and A. Shalev, Geom. Dedicata 56, No. 1, 103-113 (1995; Zbl 0836.20068)]).
In this paper, a proper subgroup \(H\) of a finite group \(G\) is said to be large if the order of \(H\) satisfies the bound \(|H|^3\leq |G|\). As the main result, the authors determine all the large maximal subgroups of finite simple groups, and establish an analogous result for simple algebraic groups (in this context, largeness is defined in terms of dimension). An application to triple factorizations of simple groups (both finite and algebraic) is discussed.
In this paper, a proper subgroup \(H\) of a finite group \(G\) is said to be large if the order of \(H\) satisfies the bound \(|H|^3\leq |G|\). As the main result, the authors determine all the large maximal subgroups of finite simple groups, and establish an analogous result for simple algebraic groups (in this context, largeness is defined in terms of dimension). An application to triple factorizations of simple groups (both finite and algebraic) is discussed.
Reviewer: Anatoli Kondrat’ev (Ekaterinburg)
MSC:
| 20D06 | Simple groups: alternating groups and groups of Lie type |
| 20E32 | Simple groups |
| 20E28 | Maximal subgroups |
| 20G15 | Linear algebraic groups over arbitrary fields |
| 20D25 | Special subgroups (Frattini, Fitting, etc.) |
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