Group Theory
Introduction
Group theory is a branch of mathematics that studies algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms.
Definition and Examples
A group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied:
1. Closure: If a and b are elements of the group, then so is a*b. 2. Associativity: If a, b, and c are elements of the group, then (a*b)*c = a*(b*c). 3. Identity element: There is an element e in the group such that, for every element a in the group, the equations e*a and a*e return a. 4. Inverse element: For each element a in the group, there exists an element b in the group such that a*b and b*a both equal the identity element.
For example, the set of integers under the operation of addition is a group. Here, the identity element is 0, and the inverse of any integer a is its negation, -a.
Historical Overview
The development of group theory sprang from three main mathematical areas: the theory of algebraic equations, number theory, and geometry. The earliest study of groups as such probably goes back to the work of the French mathematician Évariste Galois in the 1830s.
Basic Concepts and Properties
Subgroups
A subgroup of a group is a subset H of G that is itself a group, with the operation inherited from G. This is usually represented as H ≤ G.
Cosets
If H is a subgroup of a group G, and g is an element of G, then the left coset gH is the set of all products gh where h runs through H. Similarly, the right coset Hg is the set of all products hg where h runs through H.
Normal Subgroups and Quotient Groups
A subgroup N of a group G is called a normal subgroup if it is invariant under conjugation; that is, for each element n in N and each g in G, the element gng−1 is still in N. Every subgroup of index 2 is normal.
Group Representations
A representation of a group G on a vector space V over a field F is a group homomorphism from G to the group of invertible linear transformations on V. When V is of finite dimension n, this gives a homomorphism from G to the general linear group GL(n, F).
Applications of Group Theory
Group theory has many applications in physics and chemistry, particularly in the study of symmetries. The concept of a group is also central to many areas of abstract algebra, such as ring theory, field theory, and module theory.