×

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.

MSC:

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

Software:

permut; SmallGrp
PDFBibTeX XMLCite
Full Text: DOI

References:

[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
[4] Foguel, T., Groups with all cyclic subgroups conjugate-permutable groups, J. Group Theory2 (1999) 47-51. · Zbl 0921.20035
[5] Huppert, B., Über das produkt von paarweise vertauschbaren zyklischen gruppen, Math. Z.58 (1953) 243-264. · Zbl 0050.25603
[6] Xu, M. and Zhang, Q., On conjugate-permutable subgroups of a finite group, Algebra Colloq.12 (2005) 669-676. · Zbl 1079.20034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.