# Number of generators of subgroup

I am trying to prove the following.

let $$G$$ be a finitely generated abelian group, and $$H a subgroup such that there exists a subgroup $$K and we can write $$G=H \oplus K$$. Is it true that the minimal number of generators of H is strictly smaller than the minimal number of generators of $$G$$?

Clearly if G can not be written as a direct summand of $$H$$ then this is not true, just consider $$G= \mathbb{Z}$$ and $$H=2\mathbb{Z}$$.

I would like to prove it because I believe it can provide a simpler proof for the characterization of finitely generated abelian groups.

• $\mathbb Z\big / 2\mathbb Z \oplus \mathbb Z\big / 3\mathbb Z$ is cyclic. – lulu Apr 21 at 18:54
• Worth noting: "number of generators" is not well defined. I'm guessing you mean "minimal number of generators", but you should say so, – lulu Apr 21 at 18:55
• Thank you for pointing that out. I will edit to correct it. – Charles Apr 21 at 19:00

## 1 Answer

No, it is not true. Consider $$\mathbb{Z}_2\oplus\mathbb{Z}_3$$. This has a generator $$(1,1)$$. Note that $$0\oplus\mathbb{Z}_3<\mathbb{Z}_2\oplus\mathbb{Z}_3 ,$$ and $$(\mathbb{Z}_2\oplus 0)\oplus(0\oplus\mathbb{Z_3})=\mathbb{Z}_2\oplus\mathbb{Z}_3.$$ However, $$0\oplus\mathbb{Z}_3$$ is generated by $$(0,1).$$