As written in Abstract Algebra by T. W. Judson:
Lemma 13.4 : Let $G$ be a finite abelian $p-$group and suppose that $g ∈ G$ has maximal order. Then $G$ is isomorphic to $g × H$ for some subgroup $H$ of $G$.
The proof supposes that the reader already knows what maximal order means but I don't know its meaning. I searched internet and I found it either difficult/advanced to understand (e.g.) or irrelevant to specifically its meaning on a group (e.g.).
I am very new to Group Theory. Any clear simple explanation of meaning of maximal order in a group G, would be much appreciated.