What is a group $G$ of order $16$ with particular elements? I have to solve this exercise: Let $G$ a group of order $16$ such that every $g \in G$ is such that $g^2=e$. I have to determine $G$ up to isomorphisms.
We know that $G$ is abelian since $(ab)^2=e$. From the structure theorem, since the partition of $4$ are $\{4,3 1,211,1111\}$, we conclude that up to isomorphisms, $G$ is $\mathbb{Z}_{16}$, or $\mathbb{Z}_8 \times \mathbb{Z}_2$, or $\mathbb{Z}_4 \times \mathbb{Z}_2 \times \mathbb{Z}_2$, or $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$.
Is it right?

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 $\mathbb Z_{2} \oplus \mathbb Z_{2} \oplus \mathbb Z_{2} \oplus \mathbb Z_{2}$.
