Proving an Abelian Group: Understanding ( aba^{-1}b^{-1} e )

Understanding the properties of groups and specifically proving a group is Abelian is a fundamental task in group theory. This article will explore how to prove that a group G is Abelian if and only if ( aba^{-1}b^{-1} e ) for all elements a, b ∈ G. This criterion essentially restates the definition of an Abelian group, making it a valuable tool for mathematicians and enthusiasts in the field.

Definition and Basis

The problem at hand can be broken down into a clear and concise basis. In group theory, an Abelian group, or commutative group, is a group in which the operation is commutative, meaning that for all elements a, b ∈ G, the following holds:

1. ( ab ba )

This equation signifies that the order of elements does not affect the result of the group operation. Now, consider the equation ( aba^{-1}b^{-1} e ). This equation essentially claims that the product of the elements and their inverses in a specific order results in the identity element e.

Proving the Equivalence

To prove that G is an Abelian group if and only if ( aba^{-1}b^{-1} e ) for all ( a, b ∈ G ), we need to establish both directions of the proof:

1. If ( G ) is Abelian, then ( aba^{-1}b^{-1} e ) for all ( a, b ∈ G )

Let's start by assuming that G is an Abelian group. According to the definition of an Abelian group, for all a, b ∈ G, we have:

1. ab ba

Therefore, consider the expression ( aba^{-1}b^{-1} ). We can manipulate this expression as follows:

[ aba^{-1}b^{-1} aba^{-1}b^{-1} ]

Multiplying on the right by b and on the left by a^{-1}, we get:

[ a(ba^{-1}b^{-1}) (aba^{-1})b^{-1} ]

Since G is Abelian, we know that ba^{-1} a^{-1}b. Thus:

[ a(a^{-1}bb^{-1}) (aba^{-1})b^{-1} ]

[ a(a^{-1}e) (aba^{-1})b^{-1} ]

[ aea^{-1} (aba^{-1})b^{-1} ]

[ e (aba^{-1})b^{-1} ]

This shows that ( aba^{-1}b^{-1} e ) for all a, b ∈ G when G is Abelian.

2. If ( aba^{-1}b^{-1} e ) for all ( a, b ∈ G ), then ( G ) is Abelian

Now, let's assume that aba^{-1}b^{-1} e for all a, b ∈ G. We need to show that ab ba ) for all a, b ∈ G.

Starting with the given equation:

[ aba^{-1}b^{-1} e ]

Multiplying on the right by b and on the left by a^{-1}, we get:

[ (aba^{-1}b^{-1})b (a^{-1}e) ]

[ aba^{-1}b^{-1}b a^{-1} ]

[ aba^{-1} a^{-1} ]

Multiplying on the right by a, we get:

[ aba^{-1}a a^{-1}a ]

[ ab e ]

Multiplying both sides by b on the right, we get:

[ ab ba ]

This shows that G is an Abelian group if aba^{-1}b^{-1} e for all a, b ∈ G.

Hints for the Proof

If you find it challenging to come up with a proof on your own, here are some hints:

1. Starting with the equation:

a b

Try multiplying on the left or right by something to get an equation of the form (ab e).

2. Conversely:

If ab^{-1} 1, try multiplying by something to get the equation:

a b)

These hints can guide you to the necessary steps and transformations to prove the desired result.

Conclusion

To summarize, the proof that a group G is Abelian if and only if ( aba^{-1}b^{-1} e ) for all a, b ∈ G relies on the fundamental properties of group theory and can be broken down into a series of logical steps. Understanding and mastering these proofs is crucial for advancing in the field of group theory and complements your overall mathematical understanding.