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, and the Two-Step Subgroup Test, we only need to show that composition and inversion of symmetry is symmetry.

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