The order of subgroup generated by two distinct elements of order 2

Solution 1:

Your proof is correct -- $\langle x, y \rangle$ must be the entire group.

I couldnt think of any example though...

There are only two groups of order $26$ -- the cylcic group $C_{26}$, and the dihedral group $\text{Dih}_{13}$. $C_{26}$ has only one element of order $2$, so it won't be an example. So the only example is $\text{Dih}_{13}$, the group of symmetries of a $13$-gon. In this group, $x$ and $y$ are two different reflections of the $13$-gon; what you have proven is that two different reflections are enough to generate the entire set of symmetries.

In general, the groups of order $2p$ for a prime $p$ are $C_{2p}$ and $\text{Dih}_p$.