Evgeny.Makarov said:Consider $ayb=cyd$ when $y$ is the inverse of something.
Given $a$ and $b$, you can make $ayb=cyd$ true by choosing $y$, $c$ and $d$ appropriately.Alexmahone said:Even if I suppose $y=a^{-1}$ or $b^{-1}$ or $a^{-1}b^{-1}$, how do I know that $ayb=cyd$ has to be true?
An Abelian group, also known as a commutative group, is a mathematical structure consisting of a set of elements and an operation that combines any two elements to produce a third element. The operation must be associative, have an identity element, and each element must have an inverse. In an Abelian group, the operation is also commutative, meaning that the order in which the elements are combined does not affect the result.
To show that a group G is Abelian, you must demonstrate that the group satisfies the four defining properties of an Abelian group: associativity, identity element, inverses, and commutativity. This can be done by showing that the operation is associative, there exists an identity element, every element has an inverse, and the operation is commutative.
Abelian groups are important in mathematics because they have many useful properties and can be applied in various fields such as algebra, geometry, and number theory. They also serve as a foundation for more complex mathematical structures, such as rings and fields.
No, a non-Abelian group cannot become an Abelian group. The defining properties of an Abelian group must be satisfied from the beginning, and a non-Abelian group lacks the commutativity property. However, some non-Abelian groups may have subgroups that are Abelian.
One example of an Abelian group is the group of integers under addition. The operation of addition is both associative and commutative, the identity element is 0, and every integer has an inverse (its negative). This group satisfies all the defining properties of an Abelian group.