General strategies for understanding a given group presentation I know that there is no general algorithm for determining what familiar group (if any) a given presentation represents; even the problem of determining if a presentation represents the trivial group is undecidable.
However, when looking at a presentation, what are some good strategies for trying to discover truths about the group it represents? What kinds of approaches are often helpful in determining the order of a group, whether it is abelian, what kinds of subgroups it might possess, whether any of the listed generators are redundant, etc?
 A: I think the best way is to look at where the group came from. Outside of pure group theory, groups act on objects, and this is a big hint as to what the group is. It could be a group of symmetries of a geometric object, or a group of automorphisms of a space or a field. Often times, especially when dealing with free groups or things close to free groups (not too many relations), it helps to view the group as the fundamental group of a covering space, just so you can have a picture for the group.
But since it sounds like you are just working with groups for the sake of groups, here are a few tips:


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*Just start playing with the generators and relations to write down as many elements of the group as you can. Early on while learning group theory, the familiar groups are almost all finite. So you might just be able to write down every element.

*Play with just the relations to see what becomes trivial in the group. 


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*If you can generate one of the generators from the relations, that means that generator is redundant. 

*If you generate a string $aba^{-1}b^{-1}$, that means those elements $a$ and $b$ commute. 

*And there are a few common relations to look for. Like if you get the relations $a^n, b^2, (ab)^2$, there's a dihedral-like part of your group. Wikipedia has a list of other common relations, like those present in the symmetric group or in braid groups.
If you want something more formal, you can always check out Tietze transformations.
A: A good strategy is to apply the Todd-Coxeter algorithm. 
