# 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.

## Using Generators in Group Theory

• 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.

## Exploring Subgroups Generated by a Subset

• 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.

## Applications of Generators and Group Theory

• 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.