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Universal covers of finite groups. (English) Zbl 07286500
There are three well-established ways to describe a group for a computer: permutations, matrices, and presentations. Finite presentations are often a natural way to define groups. For groups given in this form, effective algorithms exist for special kinds of presentations, in general, however, due to the undecidability of the word problem for groups, many problems have been shown to be algorithmically undecidable. What one can do, based on von Dyck’s Theorem, is to attempt to investigate such a group via its quotients: this is the idea of so-called quotient algorithms.
In the article under review the authors describe how to compute finite extensions \(\widetilde{H}\) of a finite group \(H\) by a direct sum of isomorphic simple \(\mathbb{Z}_{p}H\)-modules such that \(H\) and \(\widetilde{H}\) have the same number of generators. Similar to other quotient algorithms, their description will be via a suitable covering group of \(H\). Defining this covering group requires a study of the representation module, as introduced by W. Gaschütz [Math. Nachr. 14, 249–252 (1955; Zbl 0071.25202)].
An important application of the results obtained in this paper is that they can be used to compute, for a given epimorphism \(G \rightarrow H\) and simple \(\mathbb{Z}_{p}H\)-module \(V\), the largest quotient of \(G\) that maps onto \(H\) with kernel isomorphic to a direct sum of copies of \(V\). The authors also provide a description of how to compute second cohomology groups for the group \(H\), assuming a confluent rewriting system for \(H\).
MSC:
20F99 Special aspects of infinite or finite groups
20J05 Homological methods in group theory
Software:
GAP
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References:
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