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A survey of mutually and totally permutable products in infinite groups. (English) Zbl 1026.20022

Curzio, Mario (ed.) et al., Topics in infinite groups. Rome: Aracne. Quad. Mat. 45-62 (2001).
Let \(G\) be a group, and let \(A\) and \(B\) be subgroups of \(G\). We say that \(A\) and \(B\) are ‘totally \(s\)-permutable’ if each subgroup of \(A\) is permutable with all subgroups of \(B\). The authors already described in a previous paper [J. Group Theory 2, No. 4, 377-392 (1999; Zbl 0941.20026)] the behaviour of periodic totally \(s\)-permutable subgroups. In the paper under review products of two non-periodic totally \(s\)-permutable subgroups are considered. Among other results, it is proved that if \(A\) and \(B\) are torsion-free Abelian totally \(s\)-permutable subgroups with rank \(\geq 2\), and \(A\cap B\neq\{1\}\), then the product \(AB\) is an Abelian group. Moreover, the authors obtain information on the set \(T\) consisting of all elements of finite order of \(A\cap B\), where \(A\) and \(B\) are totally \(s\)-permutable subgroups of a group \(G\). In fact, they prove that \(T\) is a hypercentral ascendant subgroup of \(AB\), all maximal primary subgroups of \(T\) are subnormal in \(AB\), and \(T\) is nilpotent of class at most \(3\) if \(AB\) is not periodic.
For the entire collection see [Zbl 0995.00010].

MSC:

20F16 Solvable groups, supersolvable groups
20E22 Extensions, wreath products, and other compositions of groups
20E15 Chains and lattices of subgroups, subnormal subgroups
20E07 Subgroup theorems; subgroup growth
20D40 Products of subgroups of abstract finite groups

Citations:

Zbl 0941.20026
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