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) $.

Thanks in advance!

  • 2
    $\begingroup$ 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: $$c\left(G/\textrm{ker}(\phi)\right)=c(G)-c(\ker(\phi)).$$ $\endgroup$ – C. Falcon Dec 27 '15 at 19:38
  • $\begingroup$ I think that actually solves my question! Thanks a lot! :) $\endgroup$ – Itay Hazan Dec 30 '15 at 15:11
  • $\begingroup$ Glad to have helped! $\endgroup$ – C. Falcon Dec 30 '15 at 16:31

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