The group $S_n$ is the group of all permutations of $n$ elements. It corresponds to the bijections of a set of $n$ elements. This group has $n!$ elements.
See here.
The subgroups of $S_n$ are called permutation groups.
The alternating group $A_n$ of order $n$ is always a normal subgroup of $S_n$.
In general, for any (infinite) set $X$, the symmetric group is denoted as $\text{Sym}(X)$.
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Author of the notes: Antonio J. Pan-Collantes
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