If $S_n$ is a symmetric group, what is the name for $n$?

Question

If $S_n$ is a symmetric group, we call $n!$ to be the size/order of a symmetric group. $S_n$ permutes $n$ symbols, what is the name for this number $n$?

Answer 1

$n$ is called the degree of the symmetric group $S_n$.

The degree of $S_n$ gives the cardinality (size) of the set whose elements are permuted by elements of $S_n$.

$S_n$ is commonly called the permutation group on $n$ "letters", (I think your use of the term "symbols" is a better choice than "letters") but is usually represented as the group of permutations of the elements in $S = \{1, 2, \cdots \, n\}, \;\;n \in \mathbb Z^+$