Permutation orders in specific symmetric groups

Question

I've recently delved into symmetric groups and stumbled upon a notation that's left me somewhat puzzled. When analyzing a particular symmetric group, let's call it $S_n$, I've come across the set notation: $\{\text{ord}(f)\ |\ f \in S_7\}$.

For context, in the case where $n=7$, would the notation "permutations of length 7" strictly refer to 7-cycle structures? Or does it encompass all potential permutations in $S_7$, such as the identity, 1-cycles, 2-cycles, and the like?

Answer 1

This is set theoretic in nature.

The notation, for a function $\varphi$ from a set $X$,

$$\{\varphi(x)\mid x\in X\}$$

means the set of values of $\varphi$ at each $x$ in $X$.

For example, for $\varphi(x)=2x$ and $X=\{1,2\}$, we get

$$\{\varphi(x)\mid x\in X\}=\{2,4\}.$$

Also ${\rm ord}: S_7\to \Bbb N$ is the order of each element of $S_7$.