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Decision problems concerning surjections and embeddings of groups

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The isomorphism problem, of deciding if two presentations define isomorphic groups, has been shown to be algorithmically impossible for finitely presented groups. However, if we start with two finite presentations P and Q which define isomorphic groups, then we can algorithmically construct an explicit isomorphism between them. Moreover, we can algorithmically enumerate all finite presentations of groups isomorphic to a given group. In this talk I will discuss variations of these ideas, when we replace the word “isomorphic” with “surjects onto” or “embeds into”, giving existing results as well as some of my own recent work.

This talk is part of the Junior Algebra and Number Theory seminar series.

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