# Abelian Group; what to do if the set is G=R-{1/3}?

• lostinmath08
In summary, the conversation discusses a problem on proving that the set G=R-{1/3} with the operation defined as x*y=x+y-3xy is an abelian group. The properties needed to prove this include the Associative Law, Neutral Element, Inverse Elements, and Commutativity. The conversation also mentions that the set G is significant in finding the solution, as the element 1/3 is the only one without an inverse under the operation and thus, including it would not make G a group.
lostinmath08

## Homework Statement

On the set G=R-{1/3} the following operation is defined:
*G: GxG arrow G

(x,y) arrow x*y=x+y-3xy

Show that (G,*) is an abelian group.

## Homework Equations

To proove something is an abelian group:

The Associative Law need to hold true x*(y*x)=(x*y)*x
Neutral Element needs to be true e*x=x*e=x for all or any x E G
Inverse Elements x*x'=x'*x=e where x' is called the inverse element of x
Commutativity x*y=y*x

## The Attempt at a Solution

I don't know how to solve this, especially with a set like this. Any Hints or Advice would be greatly appreciated

You posted a problem like this before. Commutativity should be obvious. Try solving for e. Just get started with any property you want to start out with.

I have already solved for the properties, but I thought the set matters.

The set does matter. Because once you have figured out that e=0, then 1/3 is the only element of R that doesn't have an inverse under the operation '*'. So if you include 1/3, it's not a group. Just like R-{0} is a multiplicative group and R isn't.

Last edited:

## 1. What is an Abelian group?

An Abelian group is a mathematical structure that consists of a set of elements and an operation that satisfies the properties of associativity, commutativity, identity, and inverse. It is named after the mathematician Niels Henrik Abel and is a fundamental concept in abstract algebra.

## 2. What are the properties of an Abelian group?

The properties of an Abelian group include associativity, commutativity, identity, and inverse. Associativity means that the order in which operations are performed does not affect the result. Commutativity means that the order of elements in the operation does not affect the result. Identity means that there is an element that when combined with any other element, results in the original element. Inverse means that every element in the group has an element that when combined results in the identity element.

## 3. What is the difference between an Abelian group and a non-Abelian group?

The main difference between an Abelian group and a non-Abelian group is that in an Abelian group, the operation is commutative, whereas in a non-Abelian group, the operation is not commutative. This means that the order of elements in the operation does not affect the result in an Abelian group, whereas it does in a non-Abelian group.

## 4. How can we determine if a set is an Abelian group?

To determine if a set is an Abelian group, we need to check if it satisfies the properties of associativity, commutativity, identity, and inverse. If all these properties hold true, then the set is an Abelian group. We also need to make sure that the set is closed under the operation, meaning that the result of the operation on any two elements in the set is also in the set.

## 5. What should we do if the set is G=R-{1/3}?

If the set is G=R-{1/3}, we need to determine if it satisfies the properties of an Abelian group. We can check if the set is closed under the operation, and if the operation is associative and commutative. If the set satisfies all the properties, then it is an Abelian group. If not, then it is a non-Abelian group.

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