Hungerford Chapter 2 Section 2 Problem 2 WITHOUT using the structure theorem of finite abelian groups

Let $$G$$ be a finite abelian group and $$x$$ an element of maximal order. Show that$$\langle x \rangle$$is a direct summand of $$G$$. Use this to obtain another proof of Theorem 2.1.

Theorem 2.1: Every finitely generated abelian group $$G$$ is isomorphic to a finite direct sum of cyclic groups in which the finite cyclic summands (if any) are of orders $$m_1,...,m_t$$, where $$m_1>1$$and $$m_1|m_2|...|m_t$$.

I want to show that $$G\cong \langle x \rangle \oplus G/\langle x \rangle$$. But I have difficulties in constructing the isomorphism.

The question has been answered with a method using the theorem about the structure of finite abelian groups (Finite abelian groups - direct sum of cyclic subgroup). But I don't think this exercise can be done with the theorem (or its corollaries). Thank you!

• This is not a duplicate. The suggested link would prove the structure theorem by using the structure theorem. – N. S. Apr 19 at 17:20
• @Skyshie Maybe ask again the question, and state clearly: Prove this WITHOUT using the structure theorem. – N. S. Apr 19 at 17:21
• As far as I can see the link does not use the structure theorem, but there might be more direct proofs. – Derek Holt Apr 20 at 7:20