What does $O(D)$ mean if $D$ is a congugacy class of $S_n$?

Question

I am currently reading Products of Conjugacy Classes in Groups. On page 200, it introduces some notations about the symmetric group.

Let $D\in\text{cl}(S_n)$ and $\alpha\in D$,

$k(D)=O(1)$ if $D\subseteq A_n(D\nsubseteq A_n)$

$O(D)=O(\alpha)$

I have no idea what it means. What is the meaning of having a big O value of a conjugacy class, and why we can have $D\subseteq A_n(D\nsubseteq A_n)$. I am so confused.

Answer 1

My best guess is that the second line was intended to have a zero, not a big-O. So that line defines $k(D)$ to be $0$ if $D\subseteq A_n$ and to be $1$ if $D\not\subseteq A_n$.

The third line is simply defining the notation O applied to a conjugacy class $D$ to mean $O$ of any element of that class. I would guess that $O(\alpha)$ means the order of the element $\alpha$, and so $O(D)$ means the order of any element of $D$ (which makes sense because all elements in a conjugacy class have the same order).