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An abelian group is a special type of group in mathematics. It is also known as a commutative group. This means that the order in which you perform a group operation does not matter. If you have two elements, say \( a \) and \( b \) in a group (denoted usually by \( G \)), the group is called abelian if the operation satisfies \( ab = ba \) for all elements.
Some features of abelian groups include:

• Commutative operation: As mentioned, the operation \( ab = ba \) for any elements \( a \) and \( b \).
• Closure: Performing the group operation on two elements in the group yields another element in the group.
• Associativity: When you combine more than two elements, the grouping of operations does not affect the outcome.
• Existence of identity element: There is an element \( e \) in the group such that for any element \( a \), the equation \( ae = ea = a \) holds.
• Existence of inverses: For every element \( a \) in the group, there is another element \( b \) in the group such that \( ab = ba = e \), where \( e \) is the identity element.

In the given problem, we can conclude that since \( a^2 = e \) for all \( a \), the group must be abelian, as shown through simple manipulation of group operations.

Every group has a special element called the identity element. In the context of an abelian group, this element is denoted as \( e \). Its most important property is that it does nothing when used in the group operation. For any element \( a \) in the group, combining it with the identity element \( e \) results in the same element, \( a \). In mathematical terms, this can be expressed as \( ae = ea = a \).
The identity element serves as a kind of do-nothing transformer in the world of group operations. It is the neutral element, which must be present in any group structure, whether the group is abelian or not. Its presence guarantees that each element in the group has an inverse, making it possible for every operation to be reversed.

• Invariant under operation: Combining any element \( a \) with the identity element \( e \) returns \( a \).
• Uniqueness: There is only one identity element in a group.
• Central for defining inverses: Each element \( a \) has an inverse \( a^{-1} \) such that \( aa^{-1} = a^{-1}a = e \).

In the given exercise, the identity element \( e \) helps prove that \( abab = a^2b^2 = e \), leading to the conclusion that \( ab = ba \), showing the group is abelian.

The commutative property is a fundamental characteristic of abelian groups and is what defines their nature. This property states that the order in which two elements are combined does not affect the result. So if you have two elements \( a \) and \( b \) in a group, the commutative property ensures \( ab = ba \).
This property is not just limited to addition and multiplication we learn about in elementary school math, but extends to any operation defined within a group. The commutative property ensures:

• Simplicity: Calculations become simpler because the order of operations is irrelevant.
• Consistency: Group operations behave predictably regardless of the element order.

In group theory, the commutative property has deep implications in understanding algebraic structures. In our given problem, it is used to show that \( abab = e \) can be rewritten as \( ab = ba \) with the help of the identity element, thus verifying the group is abelian.