Abelian Group Proof: Example to Show Necessity

In summary: So, you need to find a case where the identity does not hold, and then you can show that the group is not abelian.In summary, if G is an abelian group, then (ab)^2=a^2b^2 for all a, b in G.
  • #1
kathrynag
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0

Homework Statement


If G is an abelian group, then (ab)^2=a^2b^2 for all a, b in G. Give an example to show that abelian is necessary in the statement of the theorem.



Homework Equations





The Attempt at a Solution


Abelian implies commutativity.
a*b=?b*a
a^2b^2=b^2a^2
For example ab=ba must be true for the statement to work.
 
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  • #2
take for example a^2 b^2 and expand it to aabb

what can you do if ab is abelian?
 
  • #3
aabb
Implies commutativity so aabb=bbaa
I think there's something with inverse to make this work
 
  • #4
a(ab)b
 
  • #5
Ok so a(ab)b and b(ab)a
so this implies ab=ba
 
  • #6
kathrynag said:
Ok so a(ab)b and b(ab)a
so this implies ab=ba

the definition of abelian is not axb = bxa so can't do that

so instead we can swap the ab
 
  • #7
I don't really understand. You're saying we can swap the ab, but I don't gte what you mean by this.
(ab)ab=(ab)ba
 
  • #8
I don't know what waht it making you do. The point was to show that you can find G non-abelian where (ab)^2 is not equal to a^2b^2 for some a and b.
 
  • #9
Yeah and I wanted to prove that abelian implied (ab)^2=a^2b^2
Is there a hint for an a and b to use where the 2 aren't equal?
 
  • #10
(ab)^2 = a^2 b^2 --->

abab = aabb -------->

cancel the leftmost a and the rightmost b:

ba = ab
 
  • #11
well

a^2b^2 = aabb = a(ab)b

because they are abelian ab = ba

so aabb = a(ba)b = (ab)(ab)
 
  • #12
waht said:
well

a^2b^2 = aabb = a(ab)b

because they are abelian ab = ba

so aabb = a(ba)b = (ab)(ab)

But you are supposed to prove that they are Abelian.
 
  • #13
Ok I understnd how to prove it now.
I'm having trouble with the example to show abelain is necessary.
 
  • #14
kathrynag said:
Ok I understnd how to prove it now.
I'm having trouble with the example to show abelain is necessary.

It is necessary because:

(ab)^2 = a^2 b^2 --->

abab = aabb -------->

cancel the leftmost a and the rightmost b:

ba = ab

So, we have that

(ab)^2 = a^2 b^2 implies that ab = ba

This is the same as saying:

Not[ab = ba] implies Not[(ab)^2 = a^2 b^2]

Or, in plain English, if you don't have the Abelian property, you don't have that (ab)^2 = a^2 b^2. So, the Abelian property is a necessary assumption for the identity to hold.
 
  • #15
Hmmm, I think I might see this better if I saw a way where it doesn't work. For example (ab)^2 not equaling a^2b^2. Therefore, not abelian
 
  • #16
kathrynag said:
Hmmm, I think I might see this better if I saw a way where it doesn't work. For example (ab)^2 not equaling a^2b^2. Therefore, not abelian

But that's equivalent to Abelian ---> (ab)^2 =a^2b^2

which is the trivial case.

If you know that this is true (so Abelian implies the identity), then should you encounter a case where the identity is not satisfied, you know that the group cannot be Abelian.
 
  • #17
that's what I'm trying to do to find a case now where the identity is not satisfied to show then that the group is not abelian
 
  • #18
Um, as count Iblis has pointed out *any* non-abelian group will provide a counter example.
 
  • #19
kathrynag said:
that's what I'm trying to do to find a case now where the identity is not satisfied to show then that the group is not abelian

You can take some group of matrices. Matrix multiplication does not commute. So, it should be easy to find a particular example where the identity does not hold.

But such a particular example does not show that the fact that the identity does not hold implies that the group is non-abelian. To show that, you need to prove that whenever the identity does not hold, you also have that the group is not abelian.
 
  • #20
Oh, that makes sense.
 

1. What is an Abelian group?

An Abelian group is a mathematical structure consisting of a set of elements and an operation that satisfies the four properties of closure, associativity, identity, and inverse. In an Abelian group, the operation is commutative, meaning that the order in which the elements are combined does not affect the result.

2. Why is it important to prove the necessity of an Abelian group?

Proving the necessity of an Abelian group allows us to understand why the four properties mentioned above are essential for the group to function properly. It also helps to establish the fundamental principles of group theory and lays the foundation for more complex mathematical concepts.

3. Can you provide an example of an Abelian group proof?

One example of an Abelian group proof is showing the necessity of the commutative property in a group of integers under addition. This proof involves using the properties of addition and the definition of an Abelian group to demonstrate that the commutative property is necessary for the group to function as an Abelian group.

4. What is the process of proving the necessity of an Abelian group?

The process of proving the necessity of an Abelian group involves identifying the four essential properties of closure, associativity, identity, and inverse and showing how these properties are required for the group to function as an Abelian group. This is typically done through logical reasoning and mathematical equations.

5. Why is the proof of Abelian group necessary in mathematics?

The proof of Abelian group is necessary in mathematics because it helps to establish a clear understanding of the fundamental principles of group theory. It also serves as a building block for more complex mathematical concepts and provides a solid foundation for further exploration and application of group theory in various fields of mathematics and science.

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