Proving Cyclic Finite Abelian Groups of Order pn

In summary, an abelian group of order pn (where p is a prime) with p-1 elements of order p can be proven to be cyclic by ruling out certain possibilities for its structure and using the theorem for classifying finite abelian groups. By considering the number of factors of p in an element of order p in Zp^x, it can be shown that there are exactly p-1 elements of order p in the group, leading to the conclusion that it is cyclic.
  • #1
Kalinka35
50
0

Homework Statement


An abelian group has order pn (where p is a prime) and contains p-1 elements of order p. Prove that this group is cyclic.


Homework Equations





The Attempt at a Solution


I know I should use the theorem for classifying finite abelian groups, which I understand, and I feel like I have all the pieces but I don't know how to put them together.
 
Physics news on Phys.org
  • #2
Ok, so what are some possibilities for the structure of an abelian group of order p^n? Can you rule some of them out based on the order p condition?
 
  • #3
The only case I've managed to definitively rule out is the case where in
Zm1 x Zm2 x ... x Zmk we set k=n and m1=m2=...=p. In each of Zm1 through Zmk there are p-1 elements of order p so taking the direct product of all these you certainly end up with more than p-1 elements of order p.
In the case where any of m1,..., mk are equal to some p^x where x>1 and k>1 then in some Zmi there will be p^(x-1) - 1 elements with order not equal to p. I'm not sure that is entirely correct but if so then taking direct product we will end up with more than p-1 elements of order p and we'll conclude that the group has to be cyclic.
 
Last edited:
  • #4
You've got the right idea but I'm not sure you are counting correctly. How many elements of order p in Zp^x??
 
  • #5
Well, aren't the elements of order p the ones that are not divisible by p? That was what I was trying to count before. Although looking it over again I would say that there are px-px-1 elements of order p.
 
Last edited:
  • #6
An element a of order p in Zp^x satisfies the conguence a*p=p^x. How many factors of p does a have to have?
 
  • #7
Oh okay. So there are x-1 factors of p in a. So I guess this gives you that there are p-1 elements of order p since you can only multiply p^(x-1) by numbers less than p until you get p^x and the rest of the argument would be similar to the first one for setting all of the m's equal to p.

Thanks.
 

FAQ: Proving Cyclic Finite Abelian Groups of Order pn

What are Finite Abelian Groups?

Finite Abelian groups are mathematical structures that consist of a finite set of elements and an operation (usually denoted as + or *) that combines any two elements to produce a third element in the set. These groups follow the commutative property, meaning the order of operations does not affect the result.

How are Finite Abelian Groups different from other groups?

Finite Abelian groups are distinguished from other groups by their finite size and their commutative property. Other groups may have an infinite number of elements or may not follow the commutative property.

What is the significance of the term "Abelian" in Finite Abelian Groups?

The term "Abelian" comes from the mathematician Niels Henrik Abel, who proved the commutative property for finite groups in the early 19th century. It is used to describe the commutative property of these groups.

What are some examples of Finite Abelian Groups?

The simplest example of a finite Abelian group is the cyclic group, which is a group generated by a single element. Other examples include the Klein four-group, the dihedral groups, and the direct product of two cyclic groups.

What are the applications of Finite Abelian Groups?

Finite Abelian groups have applications in various fields, including cryptography, coding theory, and physics. They are also used in the study of symmetry and patterns in mathematics and can be applied to solving problems in algebra, number theory, and geometry.

Similar threads

Show that ##G\simeq \mathbb{Z}/2p\mathbb{Z}##
Replies
2
Views
1K
Group Theory: Finite Abelian Groups - An element of order
Replies
3
Views
2K
Proving that an Abelian group of order pq is isomorphic to Z_pq
Replies
5
Views
2K
Proving that Aut(K), where K is cyclic, is abelian
Replies
5
Views
1K
Abelian group as a direct product of cyclic groups
Replies
9
Views
2K
Characterizing subgroups of a cyclic group
Replies
9
Views
1K
Every infinite cyclic group has non-trivial proper subgroups
Replies
3
Views
1K
Group is a union of proper subgroups iff. it is non-cyclic
Replies
1
Views
4K
Showing a property of Abelian groups of order n
Replies
4
Views
1K
Proof that a certain group G contains a cyclic subgroup of order rs
Replies
17
Views
7K

Hot Threads

Recent Insights

Back
Top