The lower central series in some groups with the subnormal join property
HTML articles powered by AMS MathViewer
- by Howard Smith PDF
- Proc. Amer. Math. Soc. 94 (1985), 585-588 Request permission
Abstract:
The following question is considered: What groups $G$ are such that, given any subnormal subgroups $H$ and $K$ of $G$, with join $J$, and given any positive integers $a$ and $b$, there exists an integer $c$ such that ${\gamma _c}(J)$ is contained in ${\gamma _a}(H){\gamma _b}(K)$? It is shown that many, but not all, groups known to have the "subnormal join property" satisfy this further condition.References
- John C. Lennox, Daniel Segal, and Stewart E. Stonehewer, The lower central series of a join of subnormal subgroups, Math. Z. 154 (1977), no. 1, 85–89. MR 444779, DOI 10.1007/BF01215116
- John C. Lennox and Stewart E. Stonehewer, The join of two subnormal subgroups, J. London Math. Soc. (2) 22 (1980), no. 3, 460–466. MR 596324, DOI 10.1112/jlms/s2-22.3.460
- Derek S. Robinson, Joins of subnormal subgroups, Illinois J. Math. 9 (1965), 144–168. MR 170953
- Derek S. Robinson, On the theory of subnormal subgroups, Math. Z. 89 (1965), 30–51. MR 185011, DOI 10.1007/BF01111712 —, Finiteness conditions and generalised soluble groups, Vol. 2, Springer-Verlag, Berlin and New York, 1972.
- J. E. Roseblade, On groups in which every subgroup is subnormal, J. Algebra 2 (1965), 402–412. MR 193147, DOI 10.1016/0021-8693(65)90002-5
- James E. Roseblade, The derived series of a join of subnormal subgroups, Math. Z. 117 (1970), 57–69. MR 276358, DOI 10.1007/BF01109828
- Howard Smith, Commutator subgroups of a join of subnormal subgroups, Arch. Math. (Basel) 41 (1983), no. 3, 193–198. MR 721049, DOI 10.1007/BF01194828
- Howard Smith, Groups with the subnormal join property, Canad. J. Math. 37 (1985), no. 1, 1–16. MR 777035, DOI 10.4153/CJM-1985-001-8
- Stewart E. Stonehewer, Nilpotent residuals of subnormal subgroups, Math. Z. 139 (1974), 45–54. MR 352275, DOI 10.1007/BF01194143
- J. P. Williams, The join of several subnormal subgroups, Math. Proc. Cambridge Philos. Soc. 92 (1982), no. 3, 391–399. MR 677464, DOI 10.1017/S0305004100060102
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 94 (1985), 585-588
- MSC: Primary 20E15; Secondary 20F14
- DOI: https://doi.org/10.1090/S0002-9939-1985-0792265-3
- MathSciNet review: 792265