(a) Let $\theta:G→H$ be a homomorphism and let $x,y \in G$. Prove that if $x$ and $y$ are conjugate in $G$ then $\theta(x)$ and $\theta(y)$ are conjugate in $H$.
(b) By considering the homomorphism $\theta :A_5→S_5$, given by $\sigma → \sigma$, or otherwise, show that the converse of the statement in part (a) is false.
Really struggling with this question, don't know where to start - if anyone could shed some light it would be really helpful. Thanks all
Suppose $x=hyh^{-1}$. Apply $\theta$.
The second question is asking you to find two elements in $A_5$ that are conjugate in $S_5$ but not in $A_5$. Can you do this?