Group of group homomorphisms and conjugation

Question

Been stuck on this set of question for the past few days:

1. Determine the number of $S_n$-conjugacy classes of group homomorphisms from Z/3Z to $S_n$ for n = 3, 4, 5, 6.

2. Determine the number of $S_n$-conjugacy classes of group homomorphisms from Z/3Z to $S_n$ for all $n ∈ N≥1$

3. Determine the number of $GL_2(C)$-conjugacy classes of homomorphisms from Z/3Z to $GL2(C)$

Getting stuck right on definitions for the problem: I cannot find a resource online fully defining:

1. what a group of group homomorphisms is and when it is well defined
2. The definition of conjugation on such a group

Any help with tackling the actual question would also help.

Answer 1

It's not a group. The conjugation action is the action of pointwise conjugation on the target. That is, if $G, H$ are two groups, the set $\text{Hom}(G, H)$ of homomorphisms $G \to H$ acquires a conjugation action of $H$ as follows: if $h \in H$ and $\varphi : G \to H$ is a homomorphism, then $h$ acts via

$$\varphi(-) \mapsto h \varphi(-) h^{-1}.$$

The idea behind this definition is to capture which homomorphisms are related by a "change of coordinates" in the target. In the $GL_2(\mathbb{C})$ example you can think of a homomorphism $G \to GL_2(\mathbb{C})$ as a linear representation of $G$ on a vector space $\mathbb{C}^2$, and then the conjugation action identifies representations which differ by a change of basis in $\mathbb{C}^2$.

To get some sense of how this definition works, show that the conjugation action on $\text{Hom}(\mathbb{Z}, G)$ can be identified with the conjugation action of $G$ on itself. Next (and this is relevant to your exercises) show that the conjugation action on $\text{Hom}(\mathbb{Z}/n\mathbb{Z}, G)$ can be identified with the conjugation action of $G$ on its elements of order dividing $n$.