Group theory is a branch of mathematics that deals with the study of groups, which are mathematical structures that represent symmetries and transformations. In group theory, a group is defined as a set of elements with a binary operation that satisfies the properties of closure, associativity, identity, and inverse.
In an abelian group, the binary operation (usually denoted as "+") is commutative, meaning that the order in which the elements are combined does not affect the result. In a non-abelian group, the binary operation is not commutative, and the order of the elements is important in determining the result.
Some examples of non-abelian groups include the symmetric group, dihedral group, and quaternion group. The symmetric group is the group of all permutations of a set, the dihedral group is the group of symmetries of a regular polygon, and the quaternion group is a group of 8 elements that is commonly used in physics and computer graphics.
The non-abelian case of group theory has many applications in different fields, such as physics, chemistry, and computer science. In physics, non-abelian groups are used to describe the fundamental forces of nature, while in chemistry, they are used to understand the symmetries of molecules. In computer science, non-abelian groups are applied in cryptography and coding theory.
Yes, there are still many open problems and unsolved questions in group theory, especially in the non-abelian case. Some of these include the classification of all finite simple non-abelian groups, the existence of non-abelian groups with certain properties, and the study of non-abelian groups in higher dimensions.