# Conditions for size of index

For any element $$x \in A_5$$, we have that $$[A_5:C_{A_5}(x)]=\begin{cases} [S_5:C_{S_5}(x)], & \text{condition 1} \\ \frac{1}{2}[S_5:C_{S_5}(x)], & \text{condition 2} \end{cases}$$

Basically I want to know what the conditions are.

Note that $$C$$ is the centralizer.

I'm just talking to myself here, but I think condition 1 is "$$x$$ takes the form $$(abc)$$" and condition 2 is "$$x$$ takes the form $$(abcde)$$."