On groups with conjugate-permutable subgroups. (English) Zbl 1491.20054

Summary: According to T. Foguel [J. Algebra 191, No. 1, 235–239 (1997; Zbl 0920.20024)], a subgroup \(H\) of a group \(G\) is called conjugate-permutable if \(H H^x= H^xH\) for every \(x\in G \). M. Xu and Q. Zhang [Algebra Colloq. 12, No. 4, 669–676 (2005; Zbl 1079.20034)] studied finite groups with every subgroup conjugate-permutable (ECP-groups) and asked three questions about them. We gave the answers to these questions. In particular, every group of exponent 3 is an ECP-group, there exist non-regular ECP-3-groups and the class of all finite ECP-groups is neither a formation nor a variety.


20D40 Products of subgroups of abstract finite groups
20D15 Finite nilpotent groups, \(p\)-groups
20D35 Subnormal subgroups of abstract finite groups


permut; SmallGrp
Full Text: DOI


[1] A. Ballester-Bolinches, E. Cosme-Ll \(\overset{\grave{}}{\text{o}}\) pez and R. Esteban-Romero, GAP package permut — A package to deal with permutability in finite groups, v. 2.0.3, 2018.
[2] H. U. Besche, B. Eick and E. O’Brien, GAP package SmallGrp — The GAP Small Groups Library, v. 1.4.2, 2020.
[3] Foguel, T., Conjugate-permutable subgroups, J. Algebra191 (1997) 235-239. · Zbl 0920.20024
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[6] Xu, M. and Zhang, Q., On conjugate-permutable subgroups of a finite group, Algebra Colloq.12 (2005) 669-676. · Zbl 1079.20034
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