A group in group theory is a mathematical concept that represents a set of objects and a binary operation between them, where the operation follows specific rules. These rules include closure (the result of the operation must also be in the set), associativity (the order in which the operation is performed does not matter), identity (there exists an element that acts as the identity under the operation), and inverses (every element has an inverse that, when combined with the element, results in the identity).
A group is considered Abelian (or commutative) if the order in which the binary operation is performed does not affect the result. In other words, for any two elements in the group, a and b, a * b = b * a. This property is named after mathematician Niels Henrik Abel.
To prove that a group is Abelian, you must show that the group's binary operation is commutative. This can be done by showing that for any two elements in the group, a and b, a * b = b * a. This can be done using algebraic manipulation or by showing that the group follows other properties, such as the commutative property of addition or multiplication.
Proving that G is Abelian is important in group theory because it helps us understand the structure and properties of the group. Knowing that a group is Abelian can also simplify calculations and make it easier to solve problems involving the group.
No, a group cannot be both Abelian and non-Abelian. A group must follow all the properties of being Abelian to be considered as such. If even one element in the group does not follow the commutative property, the group is considered non-Abelian.