Equivalence of prime power decompositions

[ehrenfest]

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 $\mathbb{Z}_{p^{r_1}}$?

 

[ehrenfest]

anyone?

 

[ehrenfest]

please?