Permutation Group is Subgroup of Symmetric Group

Theorem
Let $S$ be a set.

Let $\struct {\Gamma \paren S, \circ}$ be the symmetric group on $S$, where $\circ$ denotes the composition operation.

Let $\struct {H, \circ}$ be a set of permutations of $S$ which forms a group under $\circ$.

Then $\struct {H, \circ}$ is a subgroup of $\struct {\Gamma \paren S, \circ}$.

Proof
Follows directly from the definition of subgroup:

$H$ is a subset of $\Gamma \paren S$, and $\struct {H, \circ}$ is a group.

Hence the result.