Group Action on Subgroup of Symmetric Group

Theorem
Let $S_n$ be the symmetric group of $n$ elements.

Let $H$ be a subgroup of $S_n$.

Let $X$ be any set with $n$ elements.

Then $H$ acts on $X$ as a group of transformations on $X$.

Proof
The identity permutation takes each element of $X$ to itself, thus fulfilling.

The product rule in $S_n$ ensures fulfilment of.

Also see

 * Stabilizer in Group of Transformations