Proving Group Abelianity Using Inverse Property

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In summary, to prove that a group G is abelian, we can use the property that whenever ab = ca, then b = c for all elements a, b, and c in G. By setting b = a^{-1}ca and c = ab(a^{-1}), we can show that b = c, thus proving that G is abelian.
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nataliemarie
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Homework Statement


Let G be a group with the following property: Whenever a,b and c belong to G and ab = ca, then b=c. Prove that G is abelian.


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The Attempt at a Solution


I started with the hypothesis ab=ca and solved for b and c using inverses. I found b=(a-1)ca and c=ab(a-1). Because the hypothesis says b=c I set them equal. (a-1)ca=ab(a-1). But I'm having trouble getting anywhere useful after that. Hints or suggestions if I'm on the right track?
 
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nataliemarie said:
I started with the hypothesis ab=ca and solved for b and c using inverses. I found b=(a-1)ca and c=ab(a-1).
You got two equations from one, so one is redundant. Just stick with [tex]b=a^{-1}ca[/tex]. Now invoke [tex]b=c[/tex], so [tex]b=a^{-1}ba[/tex]. The conclusion follows.
 

Related to Proving Group Abelianity Using Inverse Property

1. What is an abelian group?

An abelian group is a mathematical structure that satisfies the commutative property, meaning that the order in which the group elements are combined does not affect the end result.

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

To prove that a group is abelian, you must show that for any two elements in the group, their product is the same regardless of the order in which they are multiplied. This can be done by using the group's defining properties or by manipulating equations involving the group elements.

3. What are some common methods used to prove a group is abelian?

Some common methods include using the group's defining properties, such as the commutative property or the associative property, to show that the group is abelian. Another method is to use algebraic manipulations to show that the group elements commute with each other.

4. Can a group be both abelian and non-abelian?

No, a group cannot be both abelian and non-abelian. A group is either abelian or non-abelian based on whether it satisfies the commutative property or not. However, it is possible for a subgroup of a non-abelian group to be abelian.

5. Why is it important to prove that a group is abelian?

Proving that a group is abelian allows us to use the commutative property to simplify calculations and make predictions about the group's behavior. It also helps us to better understand the structure and properties of the group, which can be useful in various applications in mathematics and other fields.

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