let $|S| = 3$. Prove $(P(S),\Delta)$ is not isomorphic to $(\mathbb{Z_8},+)$.

Question

Hi everyone I hope somone can help I have some set $S$ with $3$ elements $P(S)$ is the power set.I need to prove that $(P(S),\Delta)$ is not isomorphic to $(\mathbb{Z_8},+)$. I tried building a function to show that associativity doesn't work with no success . any idea ? Thank you very much

Answer 1

$(\mathbb Z_8,+)=C_8$ is cyclic. If it would be isomorphic to $(\mathcal P(S), \Delta)$, $(\mathcal P(S), \Delta)$ would be generated by a unique element. That can’t be as $A \Delta A= \emptyset$ for all $A \in \mathcal P(S)$.