Finite abelian p-group with an element of maximal order

Question

I want to know, following theorem comes from which book?

Theorem . Suppose $G$ is a finite abelian $p$-group and $a \in G$ has maximum order, then there exists a subgroup $K⊆G$ such that:

$ \langle a\rangle+ K =G$

$\langle a\rangle\cap K =\{e\}$

Answer 1

The proof of theorem that you mentioned has two main steps :
1-Cauchy theorem : If $G$ is a finite group, and $p | |G|$ is a prime, then $G$ has an element of order $p$ .
2-If $G$ is a finite abelian $p-$group and G has a unique subgroup $H$ of order $p$, then $G$ is cyclic .
For reference you can see for example Hungerford's Agebra chapter 2 or this PDF handout.