I want to show that there is an injective homomorphism from $D_6 \to S_5$ where $D_6$ denotes the dihidral group of order 12 and $S_5$ the symmetric group. But I'm not sure how I can do this efficiently.
I define $f: D_6 \to S_5$ by $f(\sigma) = (12)$ and $f(\rho) = (123)(45)$, with $\sigma$ being a reflection and $\rho$ being a rotation.
I know that $D_6$ is generated by $\rho$ and $\sigma$ and that $S_5$ is generated $(12), (23), (34), (45)$.
So what is the fastest way, for someone who is just starting with algebra, to show that this is a homomorphism? Do I have to show it explicitly for all 12 elements?
What confuses me is that you have to show for all $x,y \in D_6$ we have $f(xy)=f(x)(y)$, while $x$ and $y$ can be any combination of $\rho$ and $\sigma$.
Lastly, what is the fastest way to show it's kernel is trivial without going over all elements?