Websites/Software for Group computation

Question

Anyone knows a website or software that helps to do computations in a group? For example, by inputting generators and relations in the group, can we tell when two particular elements in the group commute with the help of the website/software and without much tedious manual computation?

Due to unsolvability of the uniform word problem, I understand that this kind of problem cannot be solved in general. By practically speaking, can we at least solve it for some "small" or "easy" groups that frequently appears in practice?

Answer 1

This works with GAP. Here is a sample self-explanatory code (which also includes some of the other features):

gap> F := FreeGroup(2);
<free group on the generators [ f1, f2 ]>
gap> a := F.1; b := F.2;
f1
f2
gap> G := F/[a^2, b^2, (a*b)^2];
<fp group on the generators [ f1, f2 ]>
gap> H := F/[a^2, b^2, (a*b)^3];
<fp group on the generators [ f1, f2 ]>
gap> IsAbelian(G);
true
gap> IsAbelian(H);
false
gap> Comm(H.1,H.2) = Identity(H);
false
gap> StructureDescription(G);
"C2 x C2"
gap> StructureDescription(H);
"S3"

Answer 2

This is probably not what you were looking for, but anyway:

http://groupexplorer.sourceforge.net/

Answer 3

Apart from GAP, which has already been mentioned and illustrated, the other principal CAS with specialized group theory capabilities is Magma. Unlike GAP, Magma is unfortunately not open source, but there is an online calculator at http://magma.maths.usyd.edu.au/calc/ which enables you to do computations of up to 60 seconds cpu.

For your particular problem, if the groups are finite and not too big, then (unless the presentations are pathological) you can easily solve it in GAP. It becomes more interesting for infinite groups, for which there is no uniform method, but there are various possibilities for groups with specific properties.

If you have specific interestring examples in which you would like to solve these problems then please ask!