# Number of constituents in invariant factor decomposition of kernel of homomorphism

Notation. Given a finite abelian group $G$, the invariant factor decomposition theorem ensures a the existence of $k_1 \mid \cdots \mid k_n$, all different, such that $G \simeq \bigoplus_{i=1}^n \mathbb Z_{k_i}$. We denote the number of constituents in this representation by $c(G) \triangleq n$.

I'm having trouble proving the following proposition:

Proposition. Let $G$ be a finite abelian group, let $H$ be a (normal) subgroup of $G$, and let $\phi : G \to H$ be a group homomorphism. Then: $c(\ker \phi) \geq c(G) - c(H)$.

• Some thoughts: Let $\pi:G\twoheadrightarrow G/\textrm{ker}(\phi)$ be the natural surjection, there exists a unique injective group homomorphism: $$\psi:G/\textrm{ker}(\phi)\hookrightarrow H.$$ Besides, one has: $$c\left(G/\textrm{ker}(\phi)\right)=c(G)-c(\ker(\phi)).$$ – C. Falcon Dec 27 '15 at 19:38