Finite group
Definition and Basic Properties
A finite group is a group that has a finite number of elements. The number of elements is called the order of the group. Finite groups have applications in a wide range of mathematical disciplines, including number theory, algebraic geometry, and combinatorics.
Classification
The classification of finite groups is a major area of research in modern algebra. The most significant achievement in this area is the classification of finite simple groups, which states that every finite simple group belongs to one of four broad classes.
Structure
The structure of a finite group is determined by its group operation, which combines any two of its elements to form a third element. The operation must satisfy four conditions: closure, associativity, identity and invertibility.
Subgroups
A subgroup of a finite group is a subset of the group that is itself a group under the operation of the larger group. The study of subgroups is a central part of group theory.
Representations
A representation of a finite group is a way of expressing the group as a set of matrices. This allows the group's structure to be studied using linear algebra.
Applications
Finite groups have many applications in mathematics and science. They are used in the study of symmetry, in the solution of polynomial equations, and in the construction of geometric figures, among other things.