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Equivalence of prime power decompositions

  1. Mar 5, 2008 #1
    1. The problem statement, all variables and given/known data
    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

    [tex] 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 }[/tex]

    where [tex]1 \leq r_1 \leq \cdots \leq r_m[/tex] 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.

    2. Relevant equations

    3. 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 [itex]\mathbb{Z}_{p^{r_1}}[/itex] needs to generate a copy of [itex]\mathbb{Z}_{p^{r_1}}[/itex] in the group on the RHS and similarly the generator of [itex]\mathbb{Z}_{p^{s_1}} [/itex] 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 [itex]\mathbb{Z}_{p^{r_1}}[/itex]?
  2. jcsd
  3. Mar 5, 2008 #2
  4. Mar 6, 2008 #3
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