A finite symmetric group consists of the set of all transformations of a set $X$, called permutations of the elements of $X$, and in which the law of composition is the binary composition of these transformations.
Is it necessary that $X$ be the set $\{1, 2, \ldots, n\}$, or can the symmetric group of order $n$ apply to any finite set $X$ of cardinality $n$?
What about the notation $S_n$?
On Wikipedia, it is stated:
The symmetric group of degree $n$ is the symmetric group on the set $X=\{1,2,\ldots ,n\}$... If $X$ is the set $\{1,2,\ldots ,n\}$ then the name may be abbreviated to $\mathrm {S} _{n}$...
Similarly, in Peter Szekeres's "A Course in Modern Mathematical Physics" it is stated (p. 30):
The group of permutations of $X = \{1, 2, \ldots, n\}$ is called the symmetric group of order $n$, denoted $S_n$.
In Lang's "Algebra" (on page 30 in the revised 3rd edition I have), it is stated:
Let $S_n$ be the group of permutations of a set with $n$ elements. This set may be taken to be the set of integers $J_n = \{1, 2, \ldots, n\}$.
From Lang's definition, it sounds like $S_n$ can refer to any finite set $X=\{x_1, x_2, \ldots, x_n\}$ of cardinality $n$ and not necessarily the set of integers $1$ to $n$.
This makes more sense to me - I don't see why we would need to restrict it to the set of integers.
