Given a presentation of a group, how do you find what it is isomorphic to?

Question

Suppose you have a group presentation $G=<a,b|a^{5}=b^{2}=e, ba=a^{2}b>$. In general, how do you find the group it is isomorphic to?

I've seen examples using the Fundamental Theorem of Finitely Generated Abelian Groups but we haven't even covered this in class yet. Is there another procedure for finding the group to which this presentation is isomorphic?

Answer 1

In general it is difficult to decide this kind of problem. You might ponder "the group it is isomorphic to" which only makes sense if you have a canonical list of groups or a dictionary.

Here it is easier, because you have two generators and a relation which you can use to push all occurrences of $b$ to one end, so that every element is of the form $a^rb^s$ with $0\le r\le 4$ and $0\le b\le 1$. This is a general technique which is often possible with "simple" presentations - use the relations to show that every element must be of a particular form.

Now the only issue is whether there are any of these elements which are equal. Since you have ten elements and the presentation is not obviously abelian (if $a$ comes out to be equal to $e$ it will be abelian) it is natural to see if there are elements within the non-abelian group of order $10$ which satisfy these relations.

Answer 2

We can look at $G$ as a semidirect product of $<a>\cong\Bbb Z_5$ and $<b>\cong\Bbb Z_2$.

Please verify that

1. $<a>\vartriangleleft G$ (that's how I denote normal)
2. $<a><b>=G$
3. The intersection is trivial.

Notice that we know exactly what the homomorphism is, because $bab^{-1}=bab=a^2$ . Thus the group is the set $\Bbb Z _5 \times \Bbb Z _2$ with the operation $(g_1,h_1)\cdot (g_2,h_2)=(g_1 (g_2)^2,h_1h_2)$.