Groups: Generators

Understanding Generators in Group Theory

A group in mathematics is a set, equipped with an operation that combines any two of its elements to form a third element, satisfying four conditions: closure, associativity, identity and inversibility.
A generator of a group is an element of the group such that every element of the group can be expressed as the product of one or more powers of the generator.
A group generated by one element is called a cyclic group. This is because the powers of the generator create a cycle, returning to the identity after a finite number of steps.
Generators are not unique. For example, in the group of integers under addition, both 1 and -1 are generators.
Every infinite cyclic group is isomorphic (structurally identical) to the group of integers under addition. Every finite cyclic group of order n is isomorphic to the group of integers modulo n.

The order of a generator is the smallest positive integer n such that g^n is equal to the identity. For infinite cyclic groups, the generator has infinite order.
Once the generators of a group are known, one can determine the structure of the entire group. This is essential in the classification of groups.
In some contexts, it is preferred to express a group in terms of generators and relations among those generators. This is known as a presentation of the group.

The subgroup generated by a subset S of a group is the smallest subgroup of the group that contains S.
This subgroup can be constructed by taking the intersection of all subgroups of the group that contain S. Such an intersection of subgroups is again a subgroup.
The subset S consists of generators of the subgroup it generates. This is useful when studying a group through its subgroups.

Generators and relations are used in graphical and computational presentations of groups, aiding in visualising and understanding group structure and properties.
Generators are central in many areas of mathematics, including algebra, number theory, and topology.
Practical applications of group theory and its concepts like generators extend even to physics and chemistry, including crystallography, quantum mechanics, and symmetry studies.