Here is the entire question:
Let $X_{2n}=\langle x,y\mid x^n=y^2=1,xy=yx^2\rangle $ and $n=3k$ then show that $X_{2n}$ has order $6$.
Here is my question:
How do we know that $X_{2n}$ has order $6$ and not order $2$?
While tackling this question I assumed that the statement
$x^n=1$
would mean that
$x^k\neq 1$ for $1\leq k\leq n-1$
But as you go through this problem you can prove that $x^3=1$.
Well this lead to a deeper question about the uniqueness presentations.
Sure we could have the group $G=\{1,x,x^2,y,yx,yx^2\}$ satisfy the relation, but why doesn't $G=\{1,y\}$ not also satify the relation?