# Software or tool for investigating groups

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.

-

The canonical anwer is GAP.

(You can use it from inside Sage)

-
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.

-