Solve Group Theory Problem - Prime Order of G must be p^n

In summary, the conversation is discussing a problem involving finite abelian groups and primes. Specifically, it is trying to prove that the order of a group must be a power of a prime number. The conversation also brings up Lagrange's theorem and the concept of generating subgroups.
  • #1
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Hi

I have a problem I just can't seem to solve, even though the solution shouldn't be too hard

Let G be a finite abelian group and let p be a prime.
Suppose that any non-trivial element g in G has order p. Show that the order of G must be p^n for some positive integer n.

Anyone got any ideas about how to approach this??

thanks,
 
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  • #2
Suppose there is another prime q that divides the order of the group and show there must be an element of order q.
 
  • #3
but is it the case that for all factors of the order of a group there is an element of that order?? i am soo confused..
 
  • #4
You know Lagranges theorem..? Consider the subgroup generated by g,- what's his order?. Well, if you like carefully at what " generates" means, youll see that the order of the subgroup generated by g is also p.
 

1. What is group theory?

Group theory is a branch of mathematics that studies the properties of groups, which are mathematical structures consisting of a set of elements and a binary operation that combines any two elements to produce a third element.

2. What is a prime order of a group?

The prime order of a group is the number of elements in the group that cannot be broken down into smaller factors. In other words, it is the size of the group and is denoted by the symbol p^n, where p is a prime number and n is a positive integer.

3. Why does the prime order of a group have to be p^n?

This is because of Lagrange's theorem, which states that the order of any subgroup of a group must divide the order of the group. Since p^n is the only possible order for a prime group, it must be the prime order of the group.

4. How do you solve a group theory problem with a prime order of p^n?

To solve a group theory problem with a prime order of p^n, you need to first understand the basic properties of groups, such as closure, associativity, and the existence of an identity and inverse elements. Then, you can apply specific group theory theorems and techniques, such as the Sylow theorems or cyclic groups, to solve the problem.

5. What is the importance of solving group theory problems with a prime order of p^n?

Groups with a prime order of p^n have many important applications in mathematics, physics, and computer science. For example, they are used in cryptography, coding theory, and quantum physics. Solving group theory problems with a prime order of p^n helps us understand the fundamental properties of these groups and their applications in various fields.

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