Groups: Generators

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.