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.
Best Answer
In a finite group $G$, the order of an element $g\in G$ is the least positive integer $n$ such that $g^n = e$ where $e$ is the 1-element of $G$. You should prove that for each $g$ such exponents $n$ exist (this uses the assumption that $G$ is finite!), so it is possible to pick the least one, so each $g\in G$ has an order.
Now consider all the element orders. Since there are finitely many elements, we just consider a finite set of natural numbers. A finite set of numbers has a maximum. So there exists one or more group elements $g$ whose order is this maximum.
So now you know what is meant by $g$ has maximal order.
(In infinite groups, some elements may have order "infinity" (meaning no positive power of the element is $e$), but if it happens that all elements of an infinite group have a finite order, there may not be a maximum among these element orders.)