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I'm interested in software that has the ability to investigate finite groups. In particular, I'd like to be able to ask it questions like "What are the solutions to $x^3 = 1$?" (i.e. find cube roots of unity) for a permutation group.

I've had a look at Maxima but don't see a way to ask it do that.

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up vote 7 down vote accepted

The canonical anwer is GAP.

(You can use it from inside Sage)

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Thanks for your answer! Do you happen to know if GAP will solve roots of unity for finite groups? – occulus Aug 29 '14 at 19:20
You can say, for example, Filtered(SymmetricGroup(6), x -> x*x*x = ()); to get the list of elements of order three. I'm sure there are smarter ways :-) – Mariano Suárez-Alvarez Aug 29 '14 at 19:29
OOh, sold! Thanks so much! – occulus Aug 29 '14 at 19:32
In larger groups, you could start by finding conjugacy class representatives and then looking for elements of order $3$ among these. Or if the order of the group is divisible by a relatively small power of $3$, you could first find a Sylow $3$-subgroup and look for its conjugacy classes. – Derek Holt Aug 30 '14 at 16:25
Hi @DerekHolt, thanks for that info. I'm interested also in the analytical side also obviously. – occulus Sep 1 '14 at 9:09

There is also Magma, but unlike GAP it is not (in general) free.

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