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Creating all homomorphisms into a finite group

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

updated 2022-04-14 20:36:29 +0200

FrédéricC gravatar image

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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answered 2021-04-10 20:16:24 +0200

vdelecroix gravatar image

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: 220 times

Last updated: Apr 10 '21