A computer program for the calculation of a covering group of a finite group
This is a direct continuation of the same author's paper [Computational group theory, Proc. Symp., Durham/Engl. 1982, 307-319 (1984; Zbl 0544.20004)]. In the calculation of the multiplier M(P) of a p-Sylow subgroup P of a permutation group G by the NQ-algorithm actually a certain covering group \(\hat P\) of P is obtained. By methods described in the above mentioned paper, a subgroup X of M(P) can be constructed such that the p-part \(M=M(G)_ p\) of the multiplier of G is obtained as M(P)/X. Forming \(\hat P/\)X one gets a presentation of a certain extension D of M by P. D corresponds to an element \([D]\in H^ 2(P,M)\). Its image under the corestriction map \(Cor_{P,G}: H^ 2(P,M)\to H^ 2(G,M)\) is found, defining an extension \(\hat G_ p\) of M by G. Finally defining relations of a covering group \(\hat G\) of G are composed from those of the \(\hat G_ p\), and, if there are several covering groups these are constructed from the first one. The theory involved in particular in the actual construction of \(Cor_{P,G}\) is given in detail as well as a short characterization of the new programs that had to be implemented in addition to those already needed in the previous paper. The paper closes with some examples of the use of the implementation, giving presentations of the (unique) covering groups of PSL(3,4), \(M_{22}\), and PSL(4,3). (Reviewer's remark: The author has meanwhile continued this very interesting line of research by extending his methods to the calculation of cohomology groups for non-trivial modules and corresponding extensions. This will be reported about in a forthcoming paper: The mechanical computation of first and second cohomology groups.)
scientific article; zbMATH DE number 3879128