Find an injective group homomorphism $\varphi: D_4 → Sym_4$ with $D_4$ being the dihedral group

Question

Let $D_4$ be the dihedral group and $\circ$ be the composition of transformations.

Exercise:

• Find an injective group homomorphism $\varphi: D_4 → Sym_4$ and determine if $\varphi$ is an isomorphism.

Already done:

1. Show that $(D_4,\circ)$ is a group.
2. Determine $|D_4|$.

Can someone help? Thanks in advance!

Answer 1

Label the vertices of the square with the numbers $1,2,3,4$, and then corresponding to each element of $D_4$ you get a map from $\{\,1,2,3,4\,\}$ to itself, that is, a member of $S_4$. This gives you your injection (not an isomorphism, as $D_4$ has eight elements; $S_4$, $24$).