Notation. Given a finite abelian group $ G $, the invariant factor decomposition theorem ensures 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 subgroup of $ G $, and let $ \phi : G \to H $ be a group homomorphism. Then: $ c(\ker \phi) \geq c(G) - c(H) $.
Thanks in advance!