Let $G$ be an Abelian group and $p$ be a prime, $$G_p:=\{x\in G\mid x\text{ is a $p$-element }\}.$$ Then $G_p$ is a characteristic $p$-subgroup of order $|G|_p$.
My question is a simple one.
At the beginning of the proof, it was mentioned that “For $x,y ∈ G_p$ also $xy$ is a $p$-element; $\color{red}{\text{use } xy = yx}$ and 1.1.2(which is and actually should be 1.1.3 that states any algebraically closed non-empty subset is a subgroup) on page 4”.
I may think it just suffices without using $xy=yx$. What have I missed? Any help would be sincerely appreciated!
PS: It’s on page 45 of my textbook (page 58 of the pdf), The Theory of Finite Groups, An Introduction.