I'm going through "A First Course in Abstract Algebra," by Fraleigh, and I just started going through the chapters on actions on sets. I came across an exercise that asked about two groups related by a homomorphism and the implications of actions on a set:
Let $\phi:G\rightarrow H$ be a group homomorphism, and let $H$ on act a set $X$. What does this tell us about whether or not $G$ acts on $X$?
I think that it does imply that $G$ acts on $X$ (it seems rather intuitive), and I've attempted to prove it below:
If a set $H$ acts on a set $X$, then we have that:
$(i)$ $\forall x\in X$, $e_Hx=x$
$(ii)$ $(h_1h_2)(x)=h_1(h_2x)$, $\forall x\in X$, $h_1,h_2\in H$
To show $(i)$, we note that $e_H=\phi(e_G)$, which implies that:
for all $x\in X$.
To show $(ii)$, we note that by the fact that $\phi$ is a homomorphism:
for $g_1,g_2\in G$. We also have that $\phi(g_1)=h_1$, and $\phi(g_2)=h_2$. So:
by substitution into our previous statement. But the RHS is just:
Does this suffice/is there anything that can be made clearer/more concise? It seems like a fairly straightforward exercise, but the way it was phrased makes me a tad skeptical that it truly is that simple. Any advice would be sincerely appreciated. Cheers.