Show that the subgroup of $S_4=\langle (12),(13)(24)\rangle$ is isomorphic to dihedral group of order $8$ This is section 2.4 question 7 in Dummit and Foote.

Show that the subgroup of $S_4$, $A=\langle (12),(13)(24)\rangle$ is isomorphic to dihedral group of order $8$

Here $D_{4}=\langle r,s:r^4=s^2=e, srs=r^{-1}\rangle$.
I believe I want to use that $(12)(13)(24)=(1423)$ is the element mapped to by $r\in D_4$
But do I know that $\langle (12), (1423)\rangle=\langle (12), (13)(24)\rangle$?
And if I define $\phi:D_4\to S_4$ by $\phi(r)=(1432)$ and $\phi(s)=(12)$.
I can show that $\phi(s)\phi(r)\phi(s)=\phi(r)^{-1}$, since $(12)(1432)(12)=(1324)$
I believe this means that $\phi$ is a homomorphism, In Dummit and Foote they haven't proven this yet though, I'm not sure if there is some other way to prove that this $A\cong D_4$?
Also is proving a bijection enough to show that the map $\psi((12))=s, \psi((1432))=r$, is an inverse? Or do I have to show they have the same order and then show explicitly write a map for each element?
 A: *

*To show $\langle (12), (1423) \rangle = \langle (12), (13)(24)\rangle$, it suffices to check that the generators of the left-hand side belong to the right-hand side and vice versa. You've already shown the $\subseteq$ direction, so you just need to show that $(13)(24)$ can be obtained from some combination of $(12)$ and $(1423)$. (Try the first thing that comes to mind.)

*If you've checked that the relations hold, i.e. $\phi(s)\phi(r) \phi(s) = \phi(r)^{-1}$ and $\phi(r)^4 = \phi(s)^2 = e$ and you intend define to the map for other words in the natural way (e.g. $\phi(srsrs) = \phi(s) \phi(r) \phi(s) \phi(r) \phi(s)$) then you've shown that $\phi : D_4 \to S_4$ is a homomorphism.

*At this point, you've shown that $(12)$ and $(1423)$ satisfy the relations defining $D_4$. The only thing that might go wrong is that they might satisfy more relations than $D_4$. (E.g. what we've done so far would hold for a homomorphism between $D_4$ and the trivial group.) So you need to show that $\phi$ is injective to conclude. I think for this problem the easiest way to do this is to write out all the elements of $\langle (12), (1423) \rangle$ and show that it has order $8$.

