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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?

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    $\begingroup$ Take a look through the documentation for GAP. It should be able to do this (though it may require that you guarantee that the generators and relations you give produce a group with certain properties). Sage can probably also do it, but it might well be easier to go through GAP via Sage anyway. $\endgroup$ – Tobias Kildetoft Sep 9 '13 at 9:48
  • $\begingroup$ @TobiasKildetoft: Sorry for asking friend, but Do you know any codes in GAP similar to StructureDescription(...) while working with Free finitely generated semigroups? All ones below seems to leave the site. Thanks. $\endgroup$ – mrs Sep 9 '13 at 12:09
  • $\begingroup$ @BabakS. No, sorry, I have never worked with semigroups in GAP. $\endgroup$ – Tobias Kildetoft Sep 9 '13 at 12:11
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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"
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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!

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

http://groupexplorer.sourceforge.net/

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