Symmetry Group is Group

Theorem
Let $P$ be a geometric figure.

Let $S_P$ be the set of all symmetries of $P$.

Let $\circ$ denote composition of mappings.

The symmetry group $\left({S_P, \circ}\right)$ is indeed a group.

Proof
By definition, a symmetry is a bijection.

From Group of Permutations is Group, then we only need closure and inverses.

Closure follows from Composition of Symmetries is Symmetry and inverses comes from the definition of symmetry