What are the conjugacy classes in $A_5$?

Question

I'm new to group-theory and to the Alternating groups. In my book I got asked the following questions:

What are the conjugacy classes in $A_5$?

Where should I start? What does it mean, "conjugacy"?

Answer 1

The conjugacy class $[\tau]_{A_5}$ of $\tau\in A_5$ is defined by $$[\tau]_{A_5}:=\{c_\sigma(\tau)\mid \sigma\in A_5\},$$ where $$\begin{align} c_\sigma: A_5 &\to A_5, \\ \pi &\mapsto \sigma^{-1}\pi\sigma\end{align}$$ is the conjugacy map.