# Proving a particular group is abelian

• MathMike91
In summary, a group is considered abelian if its elements commute with each other under the group operation. To prove that a group is abelian, the commutativity of the group operation must be shown for all elements. A non-abelian group can be transformed into an abelian group by using the commutator operation. All subgroups of an abelian group are also abelian. Proving that a group is abelian is important in mathematics for simpler calculations and understanding of the group's structure, as well as for building important mathematical concepts and theories.
MathMike91
Thanks!

Last edited:
Think of it as

(ax)b=b(xa)

## 1. What does it mean for a group to be abelian?

A group is considered abelian if its elements commute with each other under the group operation. This means that for any two elements a and b in the group, a*b = b*a.

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

To prove that a group is abelian, you need to show that the group operation is commutative for all elements in the group. This can be done by using the group's defining properties and properties of commutativity.

## 3. Can a non-abelian group be transformed into an abelian group?

Yes, a non-abelian group can be transformed into an abelian group by adding a new operation called the commutator. The commutator measures the extent to which a group fails to be abelian and can be used to turn any group into an abelian group.

## 4. Are all subgroups of an abelian group also abelian?

Yes, all subgroups of an abelian group are also abelian. This is because a subgroup inherits the properties of the larger group, including commutativity.

## 5. Why is proving that a group is abelian important in mathematics?

Proving that a group is abelian is important in mathematics because it allows for simpler calculations and easier understanding of the group's structure. Additionally, many important mathematical concepts and theories are built upon the foundation of abelian groups.

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