Given the following group presentation $<x,y|2x+3y=0, 5x+2y=0>$ of an Abelian group, find the order of element x.
My follow up question: Is there a way to determine the order without finding some known group that the presentation is isomorphic to. I have tried to manipulate the relations and have found out 11x=0. Can I say that the order of x must divide 11?
Firstly, the free abelian group $<X,Y> \cong \mathbb{Z}^2$. Any element in $\mathbb{G}$, the group defined in the question, has the form $ax+by$, $a,b \in \mathbb{Z}$. So this question has something to do with grid, but this is not a necessary concept.
The definition relation of $\mathbb{G}$ is $$ \begin{bmatrix} 2 & 3\\ 5 & 2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} =0 $$ Note that neither row additions nor multiplying $-1$ changes the induced group. (But general scale multiplications would change the induced group). The matrix can transform to be \begin{bmatrix} -3 & 1\\ 11 & 0 \end{bmatrix} So $11x=0$ and $y = 3x$. Actually, this implies $G \cong \mathbb{Z}_{11}$.