Fixed points on higher relation modules of finite groups
Let F be a non-cyclic free group, R a non-trivial normal subgroup and set \(G=F/R\). Next let \(\gamma_ kR\) denote the k-th term of the lower central series of R. Conjugation turns the k-th factor \(R_ k=\gamma_ kR/\gamma_{k+1}R\) of this series into a G-module. The purpose of the paper is the study of the submodule \(R_ k\) G of fixed elements of \(R_ k\) for \(k\geq 1.\) Previous work of \textit{M. Auslander} and \textit{R. C. Lyndon} [Am. J. Math. 77, 929-931 (1955; Zbl 0065.259)], \textit{A. L. Shmel'kin} [Izv. Akad. Nauk SSSR, Ser. Mat. 29, 149-170 (1965; Zbl 0135.047)] and \textit{N. D. Gupta}, \textit{T. J. Laffey} and \textit{M. W. Thomson} [J. Algebra 59, 172-189 (1979; Zbl 0423.20007)] implies the following two properties of \(R\) \(G_ k\) for \(k\geq 1:\) (A) \(R\) \(G_ k\) coincides with the centre of \(F/\gamma_{k+1}R\) (B) \(R\) \(G_ k\neq \{0\}\) if, and only if, G is finite. Assume not G is finite. Then the rank of \(R\) \(G_ k\) is known thanks to results of \textit{W. Gaschütz} [Math. Z. 60, 274-286 (1954; Zbl 0056.024)] for \(k=1\), and of Gupta et al. (loc. cit.) for \(k>1\). These results amount to the computation of the O-dimensional cohomology group H \(0(G,R_ k)\). The paper under review complements them with the computation (for \(k=1)\) and estimates (for \(k>1)\) of the O-dimensional Tate cohomology group \(\hat H\) \(0(G,R_ k).\) Theorem. If \(G=F/R\) is finite then (i) \(\hat H\) 0(G,R 1)\(\cong G/G'\). (ii) The exponent of \(\hat H\) \(0(G,R_ 2)\) divides 4. (iii) The exponent of \(\hat H\) \(0(G,R_ k)\) divides k when \(k\geq 3\).
scientific article; zbMATH DE number 4031961