What is the structure of the group $\langle x, y \mid xy=yx, x^m = y^n \rangle$

Question

This group is a finitely generated Abelian group so it has a simple structure of the form

$$ \mathbb{Z} \times \mathbb{Z}_{n_1} \times \dots \times \mathbb{Z}_{n_k} $$

My question is, what is this structure? I can't find any non-trivial torsion so at this point I assume it is infinite cyclic, but I don't know how to proceed. (This is not homework)

Answer 1

This is an Abelian group. In additive notation it would be $\left<x,y\mid mx-ny=0\right>$.

The structure is given by the Smith Normal Form of the matrix $\pmatrix{m&-n}$. This is $\pmatrix{g&0}$ for $g=\gcd(m,n)$ and so $G\cong\Bbb Z/g\Bbb Z\oplus\Bbb Z$.

Answer 2

The answer depends on $m,n$.Since the group is two-generated it is a direct product $\Bbb Z\times C_k$ for some $k$. Let the infinite cyclic factor be generated by $a$ and $C_k$ by $b$. Then the generators $x,y$ satisfy $x=a^lb^p, y=a^sb^q$ where $l,s$ are co-prime and $up+vq\equiv 1 \mod k$ for some $u,v$. Since $x^m=y^n$ we have $$a^{lm} b^{pm}=a^{sn}b^{qn}.$$ So $lm=sn$ and $pm\equiv qn\mod k$. This determines $k$ and the cyclic decomposition.