What are some slick ways to prove that a presentation is actually isomorphic to a given group?

Let's say I have a particular finite presentation and want to show it's actually a presentation for the group I claim it's a presentation for. That group might be specified, say, by a linear or permutation representation, or more generally as the automorphism group of some object. (Obviously the problem of showing two different presentations are equivalent is well-studied already and I know where to look for that.)

Of course there's no general technique here that's of any use. I'm just interested in seeing nice proofs in particular cases, to get an idea of the different ways it can be done.

• What exactly are you trying to do? Are interested in a certain group and its presentation? Thanks. – mrs Feb 16 '13 at 9:59
• Look for a group $H$ that you think might be isomorphic to your group $G$ defined by a presentation, find generators of $H$ that satisfy the relations of $G$ and hence construct a homomorphism $G \to H$. Now try to find a normal form for the elements of $G$ that will help you show that you have an isomorphism. There is a good example of this being discussed right now in math.stackexchange.com/questions/305455. – Derek Holt Feb 16 '13 at 12:48
• @DerekHolt: That technique only seems to work if the presentation you're starting from has enough relators that you can just enumerate the group elements by looking at the presentation. It doesn't seem to be much help, for instance, if you want to show that $\langle a, b, c \; | \; a^2, b^3, c^5, abc \rangle \cong A_5$; even though it's obvious which elements of $A_5$ correspond to $a, b, c$ there doesn't seem to be any good normal form. In this case you could just iterate through the finitely many products and check, but of course that's no use for a countable group. – Daniel McLaury Feb 16 '13 at 23:31
• But you said that you knew that there was no general technique, and that you were looking for methods that sometimes work. The $A_5$ example is harder but it is possible, by hand, to show by coset enumeration that there are 12 cosets of the subgroup $\langle b \rangle$. You can regard coset enumeration as a systematic method of looking for a normal form. And the method is not restricted to finite groups. – Derek Holt Feb 17 '13 at 10:49
• @DanielMcLaury Im sorry, what book do You read for those about presentation of book? – Reza Habibi Jan 16 '18 at 23:41

Perhaps this isn't what you're looking for, but the question is pretty broad, so here are some general tips. When I know a presentation is supposed to describe a finite group, the first thing I usually do is try to figure out $[a,b]$ for all $a,b$ in the generating set. Then I try to find the generator orders, and from there the order of the group. Once I know that, I'll start looking for the center, derived subgroup, sylow subgroups, etc. Hopefully by that point the group will be recognizable. Sometimes using Smith normal form can help simplify the relations. For an example of this type of reasoning you might be interested in reading this answer of mine.