# Finding Presentations of Groups with GAP

Group $A_5$ has presentation $〈 a, b | a^2 = b^3 = (ab)^5 = 1 〉$. Items equal to 1 are relators, so a presentation of $A_5$ as a set of relators could be $(a^2, b^3, (ab)^5)$

$Q_{16}$ is SmallGroup(16,9) with $〈 a, b | a^4 = b^2 = abab 〉$.

More groups of size 16

In GAP, $A_5$ is SmallGroup(60,5). The following code:
RelatorsOfFpGroup(Image(IsomorphismFpGroup(SmallGroup(60,5))));
will give
[ F1^5*F2^-5, F1^5*F2^-1*F1^-1*F2^-1*F1^-1, F1^-2*F2^2*F1^-2*F2^2 ]

That's not what I'm looking for. How can I get GAP to give me a presentation, or a minimal set of relators? Is there a single line piece of code that will work for most SmallGroup items?

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You might also take a look at IsomorphismFpGroupByGenerators. – Steve D Dec 22 '11 at 20:27
There is not much to add to what Igor says, I think. There is no such thing as the presentation of a finitely presented group. GAP cannot read your mind and determine which one you want, exactly. And even if it could, then the unfortunate fact is that many problems related to finite presentation are generally algorithmically unsolvable. Such as deciding whether two given finite presentations describe isomorphic groups. – Max Horn Dec 22 '11 at 23:16

I am not sure what exactly you are looking for (GAP IS giving you some presentation). However, if you are looking for a simpler presentation, check out this discussion:

http://mail.gap-system.org/pipermail/forum/2011/003313.html

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