Groups of finite weight
HTML articles powered by AMS MathViewer
- by A. H. Rhemtulla PDF
- Proc. Amer. Math. Soc. 81 (1981), 191-192 Request permission
Abstract:
If $N$ is a group and $E$ is a group of operators on $N$ then write ${d_E}(N)$ for the minimum number of elements needed to generate $N$ as an $E$-group. It is shown that if $N$ is a normal subgroup of $E$ and $E$ acts on $N$ by conjugation, then ${d_E}(N) = {d_E}(N/Nā)$ if ${d_E}(N)$ is finite and there does not exist an infinite descending series of $E$-normal subgroups $Nā = {C_0} > {C_1} > \cdots$ with each ${C_i}/{C_{i + 1}}$ perfect. Both these conditions are, in general, necessary.References
- Reinhold Baer, Der reduzierte Rang einer Gruppe, J. Reine Angew. Math. 214(215) (1964), 146ā173 (German). MR 166267, DOI 10.1515/crll.1964.214-215.146
- K. W. Gruenberg, Free abelianised extensions of finite groups, Homological group theory (Proc. Sympos., Durham, 1977) London Math. Soc. Lecture Note Ser., vol. 36, Cambridge Univ. Press, Cambridge-New York, 1979, pp.Ā 71ā104. MR 564420
- H. Heineken and J. S. Wilson, Locally soluble groups with Min-$n$, J. Austral. Math. Soc. 17 (1974), 305ā318. Collection of articles dedicated to the memory of Hanna Neumann, VII. MR 0354874
- Michel A. Kervaire, On higher dimensional knots, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp.Ā 105ā119. MR 0178475
- P. Kutzko, On groups of finite weight, Proc. Amer. Math. Soc. 55 (1976), no.Ā 2, 279ā280. MR 399272, DOI 10.1090/S0002-9939-1976-0399272-8
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 81 (1981), 191-192
- MSC: Primary 20F05
- DOI: https://doi.org/10.1090/S0002-9939-1981-0593454-X
- MathSciNet review: 593454