Issue with definitions for conjugacy classes.

I'm trying to understand a proof about the splitting criterion for the conjugacy classes of $$A_n$$, but I think I'm getting bogged down in earlier terminology. Here's a setup.

Definitions: Let $$G$$ be a group and $$N \unlhd G$$.

1. A conjugacy class in $$G$$ is a set $$\{ghg^{-1} \mid g \in G \}$$ for fixed $$h \in G$$.
2. A $$G$$-conjugacy class in $$N$$ is a subset of $$N$$ of the form $$\{ghg^{-1} \mid g \in G\}$$ for fixed $$h \in N$$.

The proposition I'm trying to prove is the following.

Proposition: For any $$\sigma \in A_n$$ the $$S_n$$ conjugacy class of $$\sigma$$ equals the $$A_n$$ conjugacy class of $$\sigma$$ if and only if there is an odd permutation $$\tau \in S_n$$ such that $$\tau \sigma = \sigma \tau$$. If this does not hold then the $$S_n$$ conjugacy class of $$\sigma$$ is the disjoint union of two $$A_n$$ conjugacy classes; the conjugacy class of $$\sigma$$ and the conjugacy class of $$\tau\sigma\tau^{-1}$$ for any $$\tau \in S_n$$.

The proof begins as follows. Let $$C = \{g\sigma g^{-1} \mid g \in S_n \}$$. I.e $$C$$ is the conjugacy class of $$\sigma$$. Then let $$A_n$$ act on $$C$$ by conjugation, and let $$C_1,\dots,C_m$$ be the distinct orbits of this action; i.e the distinct $$A_n$$ conjugacy classes contained in $$C$$.

This last line doesn't make sense to me, but maybe it's because the definition of an $$G$$-conjugacy class didn't make sense. Why would the orbits be $$A_n$$ conjugacy classes? Because in my mind the action is $$\tau \ast a = \tau a \tau^{-1}$$ for $$\tau \in A_n$$ and $$a \in C$$, but then an orbit is just $$\text{Orb}(a) = \{\tau(g \sigma g^{-1}) \tau^{-1} \mid (g \sigma g^{-1}) \in C \}$$. But why is this an $$A_n$$ conjugacy class? To be an $$A_n$$ conjugacy class wouldn't we need I get that $$\tau$$ is an element of $$A_n$$ so we're conjugating by elements of the right set, but wouldn't we need $$g\sigma g^{-1} \in A_n$$ by the definition? Sorry I'm not sure if this makes much sense, I think I'm misunderstanding the definitions somewhere. If I need to elaborate anywhere let me know, and thanks in advance for the help.

• I'm not sure what you think is the problem. $\tau$ and $g\sigma g^{-1}$ are in $A_n$. Commented Nov 6, 2022 at 1:08
• Oh I don't know why I wasn't connecting the dots that $A_n \unlhd G$ and so $g\sigma g^{-1} \in A_n$. But then what is the proposition saying? That if there is an odd permutation $\tau \in S_n$ such that $\tau \sigma = \sigma \tau$ then $\{g \sigma g^{-1} \mid g \in S_n \} = \{g \sigma g^{-1} \mid g \in A_n\}$? Commented Nov 6, 2022 at 1:12
• That's correct. Commented Nov 6, 2022 at 1:13
• Alright, glad it was a simple oversight. If you put the comment as an answer I'll accept it to remove this from unanswered if that's recommended. Up to you. Commented Nov 6, 2022 at 1:16

Everything you have said is correct. Since $$A_n$$ is normal in $$S_n$$ and $$\sigma\in A_n$$, $$g\sigma g^{-1}$$ is in $$A_n$$, as required. And $$\tau$$ ranges over all elements of $$A_n$$, so the orbit is exactly the $$A_n$$-conjugacy class of $$g\sigma g^{-1}$$.