What is a group $G$ of order $16$ with particular elements? [duplicate]
Solution 1:
A group where all elements are idempotent is abelian. Therefore according to the theorem of classification of abelian groups, $G$ is isomorphic to a product $\mathbb Z_{p_1} \oplus \cdots \oplus \mathbb Z_{p_n}$ where $p_1, \dots, p_n$ are powers of $2$. As all elements have order equal to $2$, $G$ is isomorphic to $\mathbb Z_{2} \oplus \mathbb Z_{2} \oplus \mathbb Z_{2} \oplus \mathbb Z_{2}$.