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Infinite groups with many permutable subgroups. (English) Zbl 1175.20036

A subgroup \(H\) of a group \(G\) is permutable or quasinormal in \(G\) if \(HK=KH\) for every subgroup \(K\) of \(G\). A result of S. E. Stonehewer [Math. Z. 125, 1-16 (1972; Zbl 0219.20021)] shows that every permutable subgroup of \(G\) is ascendant in \(G\). However not every ascendant subgroup need be permutable, even in the finite case.
The groups of this paper are called AP-groups, namely groups \(G\) in which every ascendant subgroup of \(G\) is permutable in \(G\). This class of groups is very close to the class of PT-groups consisting of those groups \(G\) in which permutability is a transitive relation. Every AP-group is a PT-group but the locally dihedral group is a PT-group which is not an AP-group. Many of the very nice results in this paper are concerned with radical hyperfinite AP-groups. Among the many interesting results is the generalization of a theorem of G. Zacher [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 37, 150-154 (1964; Zbl 0136.28302)]. They give necessary and sufficient conditions for a radical hyperfinite group to be an AP-group and deduce from this that a subgroup of a radical hyperfinite AP-group is again an AP-group.

MSC:

20F99 Special aspects of infinite or finite groups
20E07 Subgroup theorems; subgroup growth
20E15 Chains and lattices of subgroups, subnormal subgroups
20E22 Extensions, wreath products, and other compositions of groups
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References:

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