The question is to prove $D\_8$ and the subgroup of $S\_4$ generated by $(1 2)$ and $(1 3)(2 4)$ are isomorphic.

I was able to show that the relations for $D\_{8}$ follow when we set $b = (1 2)(1 3)(2 4) = (1 3 2 4)$ and $a = (1 2)$ (in particular, $a^2 = b^4 = 1$, $ba = ab^{-1}$). 

I've seen solutions to this problem (and similar problems involving finite groups) that claim isomorphism follows directly from the fulfillment of the above relations, and I've seen much more involved solutions that go into the orders of the subgroups, constructing the actual homomorphism from one group to another (and then claiming it's an isomorphism), and much more.

The variance in solutions has left me confused regarding what exactly I need to show. I would love to get away with a proof as clean as claiming that isomorphism follows directly from the fulfillment of the relations. When can I do so, and why? Also, why does the fact that the relations hold imply that there exists a homomorphism from $D\_8$ to this subgroup?