Ballester-Bolinches, Adolfo; Kurdachenko, Leonid A.; Otal, Javier; Pedraza, Tatiana Infinite groups with many permutable subgroups. (English) Zbl 1175.20036 Rev. Mat. Iberoam. 24, No. 3, 745-764 (2008). 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. Reviewer: Martyn Dixon (Tuscaloosa) Cited in 1 ReviewCited in 12 Documents 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 Keywords:radical groups; hyper-\(\mathfrak X\)-groups; AP-groups; PT-groups; permutable subgroups; quasinormal subgroups; ascendant subgroups; hyperfinite groups Citations:Zbl 0219.20021; Zbl 0136.28302 × Cite Format Result Cite Review PDF Full Text: DOI Euclid EuDML References: [1] Alejandre, M. J., Ballester-Bolinches, A. and Pedraza-Aguilera, M. C.: Finite soluble groups with permutable subnormal subgroups. J. Algebra 240 (2001), 705-722. · Zbl 0983.20014 · doi:10.1006/jabr.2001.8732 [2] Ballester-Bolinches, A., Esteban-Romero, R. and Pedraza-Aguilera, M. C.: On finite groups in which subnormal subgroups satisfy certain permutability conditions. In Advances in Algebra , 38-45. World Sci. 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