I am trying to find out what the following group is:
$$G = \langle a, b \mid ab^2 = b^2a,\ a^4 = b^3\rangle.$$
Due to the isomorphism problem for groups, there is no algorithmic way to approach questions like this in general. The only technique I know of is to consider the abelianisation of $G$, which, if I'm not mistaken, is the group given by the same generators and relations, together with the additional relation $ab = ba$. In this case, aside from the fact that you can remove the first relation, it doesn't seem be any simpler to determine.
So my questions are as follows:
- What is the group $G$?
- Without using the answer to the first question (i.e. using only the presentation), what is the abelianisation of $G$?
In addition, any descriptions of other techniques that one can use to get a better understanding of a group from a presentation would be very much appreciated.