Group Action on Subgroup of Symmetric Group

Theorem
Consider the Symmetric Group $$S_n$$.

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 GA-1.

The product rule in $$S_n$$ ensures fulfilment of GA-2.