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 $\struct {S_P, \circ}$ is indeed a group.

Proof
By definition, a symmetry mapping is a bijection, and hence a permutation.

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

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