Proving o(G) Divisible by p in Abelian Groups

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In summary, the conversation discusses a problem involving an abelian group and a prime number. The goal is to show that G(p) is equal to the set of elements in G with an order of p^k for some k. The conversation includes speculation on the meaning of G(p) and mentions the use of Cauchy's Theorem. The original poster later clarifies that G(p) was not defined properly, but they were able to solve the problem using the fundamental theorem of arithmetic.
  • #1
Singularity
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Hey guys. I've been stuck on the following thing for a little while now. Some help would be appreciated.

If p divides o(G) (G an abelian group and p a prime), then show that
G(p) = {g from G | o(g) = p^k for some k }

I keep going round in circles.

P.S. - this is not a homework question, just something I saw in an abstract algebra book that they stated without proof.
 
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  • #2
Singularity said:
Hey guys. I've been stuck on the following thing for a little while now. Some help would be appreciated.

If p divides o(G) (G an abelian group and p a prime), then show that
G(p) = {g from G | o(g) = p^k for some k }

I keep going round in circles.

P.S. - this is not a homework question, just something I saw in an abstract algebra book that they stated without proof.

What is G(p)? Are you sure this wasn't a definition?
 
  • #3
What about Cauchy's Theorem?
 
  • #4
Singularity said:
If p divides o(G) (G an abelian group and p a prime), then show that
G(p) = {g from G | o(g) = p^k for some k }

If you don't define G(p) for us we can't help. (The sylow p subgroup perhaps? but that is an easy exercise - all elements of order p^k lie in some sylow subgroup, and there is only one in an abelian group - so it must be something more difficult than that.)
 
Last edited:
  • #5
Did it:)

Thanks to all who posted. My apologies for not defining G(p) properly. In any event, I solved the problem. The fundamental theorem of arithmetic did the trick (I underestimated the strength of the uniqueness of prime factorization in my earlier attempts!). Ciao
 

What is the definition of "proving o(G) divisible by p in Abelian Groups"?

The term "proving o(G) divisible by p in Abelian Groups" refers to showing that the order (number of elements) of an Abelian group G is divisible by a prime number p. In other words, there exists a positive integer k such that o(G) = kp.

Why is proving o(G) divisible by p important in Abelian Groups?

This is important because it helps us understand the structure of the Abelian group better. It also has implications in other areas of mathematics, such as group theory and number theory.

What are some strategies for proving o(G) divisible by p in Abelian Groups?

One strategy is to use the Fundamental Theorem of Finite Abelian Groups, which states that every finite Abelian group can be written as a direct product of cyclic groups. Another strategy is to use the Sylow theorems, which provide conditions for the existence of subgroups of certain orders in a group.

Can o(G) be divisible by more than one prime number in Abelian Groups?

Yes, o(G) can be divisible by multiple prime numbers in Abelian Groups. For example, if o(G) = p*q, where p and q are distinct prime numbers, then o(G) is divisible by both p and q.

Are there any real-life applications of proving o(G) divisible by p in Abelian Groups?

While this concept may not have direct real-life applications, it is a fundamental concept in group theory and has implications in other areas of mathematics such as cryptography and coding theory. It also helps us understand the properties and structure of Abelian groups, which can be applied in various fields of science and engineering.

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