Let $G$ be a group such that $a^2=e$ for all $a \in G$. Show that $G$ is isomorphic to $\mathbb{Z}_2\times\cdots\times \mathbb{Z}_2$ ($n$ factors).

Question

I aim to show the following fact: Let $G$ be a group such that $a^2=e$ for all $a \in G$. Show that $|G|=2^n$ and $G$ is isomorphic to $\mathbb{Z}_2\times\cdots\times \mathbb{Z}_2$ ($n$ factors).

I can prove that G is a $p$-group, with $|G|=2^n$. What theorem or fact could guarantee that isomorphism?

Answer 1

@πr8 has shown $G$ is Abelian. You are given $|G|$ is finite, so $G$ is finitely generated. Therefore, $G$ is a finitely generated Abelian group, about which there is a big theorem. (It is related to the Fundamental Theorem of Finitely Generated Modules over a PID.) Finally, I observe that $2$ is in the annihilator of the $\Bbb{Z}$-module $G$, which with the big theorem, leaves no options for the factor structure of $G$.

Answer 2

Hint: show that $G$ is abelian and has structure of a vector space over $\mathbb Z_2$.