an:01028964
Zbl 0885.20018
Brown, Ronald; Porter, Timothy
On the Schreier theory of non-Abelian extensions: generalisations and computations
EN
Proc. R. Ir. Acad., Sect. A 96, No. 2, 213-227 (1996).
00041739
1996
j
20E22 20F05 20J05 57M20 18G40
crossed modules; identities among relations; group extensions with non-Abelian kernels; crossed sequences; crossed complexes; free crossed resolutions
The paper addresses the classification of group extensions \(1\to A\to E\to G\to 1\) of a group \(A\) by a group \(G\) with an emphasis on the case where \(A\) is non-abelian and offers various generalizations thereof. Using the notions of crossed module and of the module of identities among relations [cf. e. g. \textit{R. Brown} and the reviewer, Lond. Math. Soc. Lect. Note Ser. 48, 153-202 (1982; Zbl 0485.57001)], the authors of the paper under review give a modern version and a generalization of results of \textit{A. M. Turing} [Compos. Math. 5, 357-367 (1938; Zbl 0018.39201)] (which rely on earlier results of Schreier and Reidemeister). In particular, they examine in detail the construction that results when the transcription of the usual 2-cocycle condition fails. This leads to a crossed sequence rather than to an ordinary group extension, and results on the classification of crossed sequences are given as well. The main result, Theorem 1.2, somewhat provides a framework for computations. In fact, the use of the crossed complex theory gives an easy access to finitary computations provided a suitable small free crossed resolution is available. This is illustrated with the standard presentation of the trefoil group \(G\) and with other examples.
Reviewer's remark: Related relevant references (which are not given) are the reviewer's two papers [J. Reine Angew. Math. 321, 150-172 (1981; Zbl 0441.20033) and J. Algebra 72, 296-334 (1981; Zbl 0462.18008)].
J.Huebschmann (Villeneuve d'Ascq)
Zbl 0485.57001; Zbl 0018.39201; Zbl 0441.20033; Zbl 0462.18008