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Maximal subgroups of direct products. (English) Zbl 0888.20016
The article under review describes maximal subgroups of the direct product \(G^n\) of \(n\) copies of a group \(G\). The reason for this investigation is a well known criterion for simplicity of a group \(G\) for which the diagonal subgroup of \(G\times G\) is a maximal subgroup. In particular, as shown in this article, if \(G\) is perfect then any maximal subgroup of \(G^n\) is the inverse image of a maximal subgroup of \(G^2\) for some projection \(G^n\to G^2\) onto two factors, and if \(G\) is perfect and finite then the number of maximal subgroups of \(G^n\) is a quadratic function of \(n\). The author also deduces a theorem of Wiegold about the growth behavior of the number of generators of \(G^n\).

20E28 Maximal subgroups
20E32 Simple groups
20F05 Generators, relations, and presentations of groups
20E22 Extensions, wreath products, and other compositions of groups
20E36 Automorphisms of infinite groups
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