- #1
ehrenfest
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Homework Statement
Let G be a finitely generated abelian group and let T_p be the subgroup of all elements having order some power of a prime p. Suppose
T_p \simeq \mathbb{Z}_{p^{r_1}} \times \mathbb{Z}_{p^{r_2}} \times \cdots \times \mathbb{Z}_{p^{r_m }} \simeq \mathbb{Z}_{p^{s_1}} \times \mathbb{Z}_{p^{s_2}} \times \cdots \times \mathbb{Z}_{p^{s_m }
where 1 \leq r_1 \leq \cdots \leq r_m and similarly for the s_i.
Prove that n = m and s_i = r_i for all i. Hint: first prove r_1 = s_1 and then use induction.
Homework Equations
The Attempt at a Solution
I can prove that n = m.
Let phi be the isomorphism from the middle group to the right hand group in the line above. Under phi, the generator of \mathbb{Z}_{p^{r_1}} needs to generate a copy of \mathbb{Z}_{p^{r_1}} in the group on the RHS and similarly the generator of \mathbb{Z}_{p^{s_1}} needs to generate a copy of that group under phi^{-1}, but that doesn't really prove that r_1 = s_1. How can you really use the fact that r_i and s_i are minimal? How can you say anything about the factorization just knowing that e.g the right hand group contains a copy of \mathbb{Z}_{p^{r_1}}?