Symmetric Difference on Power Set forms Abelian Group

Theorem
Let $S$ be a set such that $\O \subset S$ (that is, $S$ is non-empty).

Let $A \symdif B$ be defined as the symmetric difference between $A$ and $B$.

Let $\powerset S$ denote the power set of $S$.

Then the algebraic structure $\struct {\powerset S, \symdif}$ is an abelian group.

Proof
From Power Set is Closed under Symmetric Difference, we have that $\struct {\powerset S, \symdif}$ is closed.

The result follows directly from Set System Closed under Symmetric Difference is Abelian Group.