Symmetric Difference on Power Set forms Abelian Group

Theorem

Let $S$ be a set such that $\varnothing \subset S$ (i.e. $S$ is not empty).

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

Let $\mathcal P \left({S}\right)$ be the power set of $S$.

Then the algebraic structure $\left({\mathcal P \left({S}\right), *}\right)$ is an abelian group.

Proof

From Power Set Closed under Symmetric Difference, we have that $\left({\mathcal P \left({S}\right), *}\right)$ is closed.

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

$\blacksquare$

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): Exercise $4.12$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 7$: Example $7.4$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): Exercise $7.1 \ \text{(b)}$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967)... (previous)... (next): $\text{II}$: Exercise $\text{T}$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 26 \kappa$