We concider two groups:
$(P(X),\Delta)$ - group of power set of $X$ with operation defined as the symmetric difference, $|X|=n$.
the group of n-tuples $(a_1,\ldots ,a_n)$ (where $a_i= 1$ or $a_i=0$ for all $i$) with the operation $(a_1,\ldots ,a_n)\oplus(b_1,\ldots ,b_n)=(a_1\oplus b_1,\ldots ,a_n\oplus b_n)$
Both groups have the same order and my question is if there is such an isomorphism between them? If so, how to construct it?
The natural candidate for an isomorphism is $X \mapsto ([x_1 \in X], \dots, [x_n \in X])$ where $X = \{ x_1, \dots, x_n \}$ and $[x_i \in X]=1$ or $0$ according to whether $x_i \in X$.