Prove: a finite abelian group $G$ in which each element has order 1 or $p$, $p$ prime, is an elementary abelian group.
Definition: Let $p$ be a prime number and $n\ge 1$. Then $E_{p^n}:=C_p \times C_p \times\cdots\times C_p$ is an elementary abelian group of order $p^n$.
My attempt:
Let $n=|G|$. Consider $g\ne e \in G$, then $\langle g \rangle \cong C_p \trianglelefteq G$ since $G$ is given to be abelian. Choose $h\ne g\ne e\in G$, and let $x\in \langle g\rangle \cap \langle h\rangle$. Element $x$ has order $p$, thus $\langle x \rangle = \langle g\rangle \cap \langle h\rangle \cong C_p$.
I don't see how I can conclude that $\langle g\rangle \cap \langle h\rangle$ is trivial. Then I'd also have to show that $\prod_{g\in G} \langle g\rangle = G$, to find that $G\cong C_p \times \cdots\times C_p$.
Any ideas?
Thanks.
Best Answer
Choose $h\notin \langle g\rangle$. Then, by Lagrange's theorem, $\langle g\rangle \cap \langle h\rangle$ is trivial.
If $G$ is finite, then the process of repeatedly choosing elements not in the product that you have at that stage must stop and will give you the direct product you require.