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I want to prove , using the fundamental theorem for finitely generated abelian groups, that if $G$ is a finitely generated abelian group and $G_t$ is its torsion subgroup such that $G/G_t$ is of rank $m$ and $H$ is a subgroup such that $H/H_t$ is of rank $n$, then $(G/H)/(G/H)_t$ is of rank $m-n$. There is a solution Rank of $(G/H)/(G/H)_t$ where $G$ is finitely generated abelian and $H$ is a subgroup. but its proof uses notions with which I am not familiar.

I could solve it by assuming that every subgroup of $G$ is a direct sum of factors each of which is a subgroup of the corresponding factors in the decomposition of $G$, but this is not true in general. What should I do?

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  • $\begingroup$ Why do you assume that everyone will know what $G_t$ means? $\endgroup$
    – Derek Holt
    Jun 2, 2018 at 16:53
  • $\begingroup$ I thought it was a standard notation for the torsion subgroup. I’ll edit my question @DerekHolt $\endgroup$
    – user555729
    Jun 2, 2018 at 16:57

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