Order of the group.

Question

An abelian group $G$ is generated by $x$ and $y$ with $$O(x)=16,O(y)=24,x^2=y^3$$ What is the order of $G$?

My attempt:There are $24+16-1=39$ elements generated by $x$ and $y$ separately. Also $xy=yx\in G$

And $x^2y=y^4$ which is already counted.

$x^3y=xy^4\in G$

Similarly I found that $xy^7,xy^{10},xy^{13},xy^{16},xy^{19},xy^{22}\in G$

This becomes very difficult and lengthy.

If you have some other quick method to find the order please help me.Also give the correct answer,I do not know the answer.

Thanks.

Answer 1

hint $x^{2n}=y^{3n}$ and $x^{2n+1}=xy^{3n}$ so factoring out the subgroup $Y=<y>$ gives only two cosets

Answer 2

Since $|x|=16$ and $|y|=24$, we cannot have $x \in \langle y \rangle$, but $x^2=y^3 \Rightarrow x^2 \in \langle y \rangle$, so $|G|=2|\langle y \rangle|=48$.