×

zbMATH — the first resource for mathematics

On the cyclic subgroup separability of the free product of two groups with commuting subgroups. (English) Zbl 1308.20031
Author’s summary: Let \(G\) be the free product of groups \(A\) and \(B\) with commuting subgroups \(H\leq A\) and \(K\leq B\), and let \(\mathcal C\) be the class of all finite groups or the class of all finite \(p\)-groups. We derive the description of all \(\mathcal C\)-separable cyclic subgroups of \(G\) provided this group is residually a \(\mathcal C\)-group. We prove, in particular, that if \(A,B\) are finitely generated nilpotent groups and \(H,K\) are \(p'\)-isolated in the free factors, then all \(p'\)-isolated cyclic subgroups of \(G\) are separable in the class of all finite \(p\)-groups. The same statement is true provided \(A,B\) are free and \(H,K\) are \(p'\)-isolated and cyclic.

MSC:
20E26 Residual properties and generalizations; residually finite groups
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1007/BFb0058924
[2] Bezverhnii V. N., Izv. Tulsk. Gos. Un-ta, Ser. Mathematics 7 pp 21– (2001)
[3] DOI: 10.1112/plms/s3-7.1.29 · Zbl 0077.02901
[4] DOI: 10.1007/BF01434960 · Zbl 0308.20027
[5] DOI: 10.1090/S0002-9947-1970-0260879-9
[6] DOI: 10.1006/jabr.1997.7256 · Zbl 0918.20009
[7] Loginova E. D., Nauch. Tr. Ivan. Gos. Univ. Math. 3 pp 49– (2000)
[8] Magnus W., Combinatorial Group Theory (1966)
[9] Novikova O. A., Chebyshevskii Sb. 2 pp 73– (2001)
[10] Sokolov E. V., Lobachevskii J. Math. 11 pp 27– (2002)
[11] Sokolov E. V., Separability of Subgroups by Certain Classes of Groups (2012)
[12] DOI: 10.1155/IJMMS.2005.2287 · Zbl 1091.20021
[13] DOI: 10.1080/00927870600875906 · Zbl 1114.20019
[14] DOI: 10.1155/2010/803243 · Zbl 1205.20040
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.