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# Creating all homomorphisms into a finite group

Let $G=< g_1,\ldots ,g_n| r_1,\ldots r_k>$ be a finite presentation of a group and let $S$ be a finite group, let's say for concreteness the symmetric group on 4 elements.

We want to create all homomorphisms $G\rightarrow S$.

One could just do the brute force search by looking at all maps from the set of generators $g_1,\ldots, g_n$ to $S$ and check which ones are homomorphisms.

Is this (or a more clever way of doing this) already implemented?

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Depending on G, you can not even check whether two homomorphisms are equal or not (due to the unsolvability of the word problem).

If you want specifically S = SymmetricGroup(4) then homomorphisms G -> S are in correspondence with the stabilizer of 1. So this is the same thing as looking at subgroups of index 4 in your group G. If your group is "nice enough" that should be faster than brute force.

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Asked: 2021-04-02 13:57:46 +0200

Seen: 39 times

Last updated: Apr 10