Maximal subgroups of small index of finite almost simple groups.(English)Zbl 1509.20043
All groups under consideration are finite.
An almost simple group \(R\) is a subgroup of \(\operatorname{Aut}(S)\) for some simple group \(S\), such that \(S\leq R\). If \(S\) is a non-abelian simple group, then the socle of \(R\) is \(S\). Let \(l(X)\) be the smallest index of a core-free subgroup of a group \(X\).
In the paper under review, the authors prove that every almost simple group \(R\) with socle isomorphic to a simple group \(S\) possesses a conjugacy class of core-free maximal subgroups whose index coincides with the smallest index \(l(S)\) of a maximal subgroup of \(S\) or a conjugacy class of core-free maximal subgroups with a fixed index \(v_S\leq l(S)^2\), depending only on \(S\).
In addition, it is argued that if \(S\) is a non-abelian simple group, then \(l(S)^2<|S|\) and the number of subgroups of the outer automorphism group of \(S\) is bounded by \(\log_2^3 l(S)\).
All results largely depend on the classification of finite simple groups.
An almost simple group \(R\) is a subgroup of \(\operatorname{Aut}(S)\) for some simple group \(S\), such that \(S\leq R\). If \(S\) is a non-abelian simple group, then the socle of \(R\) is \(S\). Let \(l(X)\) be the smallest index of a core-free subgroup of a group \(X\).
In the paper under review, the authors prove that every almost simple group \(R\) with socle isomorphic to a simple group \(S\) possesses a conjugacy class of core-free maximal subgroups whose index coincides with the smallest index \(l(S)\) of a maximal subgroup of \(S\) or a conjugacy class of core-free maximal subgroups with a fixed index \(v_S\leq l(S)^2\), depending only on \(S\).
In addition, it is argued that if \(S\) is a non-abelian simple group, then \(l(S)^2<|S|\) and the number of subgroups of the outer automorphism group of \(S\) is bounded by \(\log_2^3 l(S)\).
All results largely depend on the classification of finite simple groups.
Reviewer: Mikhail Kabenyuk (Kemerovo)
MSC:
| 20E28 | Maximal subgroups |
| 20D05 | Finite simple groups and their classification |
| 20B15 | Primitive groups |
Software:
ATLAS Group RepresentationsCite
References:
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