# Equivalence of prime power decompositions

• ehrenfest
In summary, the conversation discusses proving that the number of factors and the powers of the prime in the subgroups of a finitely generated abelian group are equal. The hint is to use induction and show that the generators of the subgroups must generate copies of themselves under the isomorphism, but further clarification is needed on how to use the minimality of the powers.
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.

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

anyone?

## 1. What is the equivalence of prime power decompositions?

The equivalence of prime power decompositions is a concept in number theory that states that two positive integers are equivalent if they can be expressed as a product of the same set of prime numbers, raised to the same powers.

## 2. How is the equivalence of prime power decompositions useful in mathematics?

The equivalence of prime power decompositions is useful in mathematics for simplifying calculations and identifying patterns in numbers. It also plays a crucial role in the study of algebraic structures and prime factorization.

## 3. Can two different sets of prime numbers have the same prime power decomposition?

No, two different sets of prime numbers cannot have the same prime power decomposition. This is because the prime factorization of a number is unique, and each prime number can only be used once in the decomposition.

## 4. How does the equivalence of prime power decompositions relate to the Fundamental Theorem of Arithmetic?

The Fundamental Theorem of Arithmetic states that every positive integer can be expressed as a unique product of primes. This is closely related to the equivalence of prime power decompositions, as it also shows the uniqueness of prime factorization.

## 5. Can the equivalence of prime power decompositions be extended to other mathematical structures?

Yes, the concept of equivalence of prime power decompositions can be extended to other mathematical structures, such as polynomials and matrices. In these cases, the prime numbers are replaced by irreducible elements or irreducible matrices, respectively.

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