Need help proving a group is abelian

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In summary, the task at hand is to prove that any group of order 9 is abelian. One possible approach is to create a Cayley table, but for a group of nine elements, this may not be the most efficient method. Alternatively, if the group has an element of order 9, it is automatically abelian. If not, all non-identity elements must have order 3, which can be used to prove the group's abelian nature.
  • #1
vince72386
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I have a midterm tomorrow morning and I am completely lost on how to finish the problem, I was told a question tomorrow will mirror this one so any help is appreciated.

Question:

Prove any group of order 9 is abelian.


Answer:

Let G be a group such that |G|=9

One of these elements has to be the identity.

The remaining 8 will consist of 4 elements and their respective inverses.


Where do I go from here?
 
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  • #2
perhaps making a cayley table will help.
 
  • #3
How do I go about creating a cayley table?
 
  • #4
You put all the elements, presumable a, b, c, d, e, f, g, h, i in a table, across and down (like a multiplication table) and then fill in a*a=? a*b=?

But for nine elements this may not be the best way to approach this problem.
 
  • #5
What is another way of approaching it without constructing the tables?
 
  • #6
If |G|=9, then if G has an element of order 9, then it's a cyclic group and it's abelian. Problem solved. If not then all nonidentity elements of G must have order 3, right? Start from there.
 

FAQ: Need help proving a group is abelian

What is an abelian group?

An abelian group is a mathematical structure that follows the commutative property. This means that the order of elements in the group does not affect the outcome of operations. In other words, for any two elements a and b in the group, a * b = b * a.

How do I prove that a group is abelian?

To prove that a group is abelian, you must show that it satisfies the commutative property for all elements in the group. This can be done by demonstrating that the group's operation is commutative, or by showing that its elements can be rearranged in a way that satisfies the commutative property.

What are some common examples of abelian groups?

The most well-known example of an abelian group is the set of real numbers with addition as the operation. Other common examples include the set of integers with addition, the set of rational numbers with addition, and the set of complex numbers with addition.

What are the benefits of proving a group is abelian?

Proving that a group is abelian can help simplify calculations and make the group easier to work with. It also allows for the use of specific theorems and properties that only apply to abelian groups, making it a useful tool in mathematical proofs.

Are there any strategies or tips for proving a group is abelian?

One strategy for proving a group is abelian is to use the definition of an abelian group and check if it applies to the given group. Another approach is to look for patterns in the group's elements and operations, and see if they follow the commutative property. It can also be helpful to try manipulating the elements in different ways to see if they still satisfy the commutative property.

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