Proving Abelian Group with Numbers and Operations

In summary, the conversation discusses how to prove if a set of numbers with an arbitrary operation is an abelian group. The first step is proving that it is a group, and then looking at the definition of an abelian group. The hint is given to test the commutativity property, and the conversation ends with a sarcastic comment about people repeating what is already written.
  • #1
TimNguyen
80
0
Hello.

I was wondering how I could prove if a set of numbers along with some arbitrary operation is an abelian group.
 
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  • #2
Don't you have a list of axioms which says "G is an abelian group if and only if the following are satisfied: ..."?
 
  • #3
The first step is proving that it is a group in the first place.

If it's not a group, than it certainly isn't an abelian group.

Then after that, look at the definition of what it means for a group to be abelian. Test it. Then you are done.

Hint: a*b = b*a (where * is the binary operation)
 
  • #4
Sometimes it feels like people just repeat what I write.
 
  • #5
Muzza said:
Sometimes it feels like people just repeat what I write.

You said...

Don't you have a list of axioms which says "G is an abelian group if and only if the following are satisfied: ..."?

That's assuming that it is a group, but he hasn't even shown that yet. I'd worry about proving that it is a group before even thinking about what properties the group might have.
 
  • #6
That's assuming that it is a group,

No it isn't. I even used the plural of "axiom" to try to hint at the fact that several things, rather than just commutativity, needed to be checked.
 
  • #7
Muzza said:
No it isn't. I even used the plural of "axiom" to try to hint at the fact that several things, rather than just commutativity, needed to be checked.

Good hint. :rolleyes:
 

1. What is an Abelian group?

An Abelian group is a mathematical structure that consists of a set of elements and an operation that combines any two elements to form a third element, following certain rules. These rules include closure, associativity, identity, and invertibility. In an Abelian group, the operation is commutative, meaning the order in which the elements are combined does not affect the result.

2. How do you prove that a group is Abelian?

To prove that a group is Abelian, you need to show that the operation is commutative, meaning that a*b = b*a for all elements a and b in the group. This can be done by using the group's definition and properties, such as closure, associativity, and identity. If the group satisfies all of these properties, then it can be proven to be Abelian.

3. Can a set of numbers and operations form an Abelian group?

Yes, a set of numbers and operations can form an Abelian group if they satisfy the four group properties: closure, associativity, identity, and invertibility. For example, the set of integers with addition as the operation form an Abelian group.

4. How does the commutative property relate to an Abelian group?

The commutative property is a crucial aspect of an Abelian group. It states that the order in which the elements are combined does not affect the result. In other words, the operation is commutative. This property allows for the group to be symmetric, which is a defining characteristic of an Abelian group.

5. What are some examples of Abelian groups?

Some common examples of Abelian groups include the set of integers with addition, the set of non-zero rational numbers with multiplication, and the set of complex numbers with addition. Other examples include the set of real numbers with multiplication, the set of even numbers with addition, and the set of positive integers with multiplication.

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