Presentation of a group

Presentation of a Group

The presentation of a group is a method of defining a group in terms of a set of generators and a set of relations among those generators. This approach is particularly useful in abstract algebra and group theory for describing groups in a compact form.

Definition

A presentation of a group \( G \) is given by: \[ G = \langle S \mid R \rangle \] where \( S \) is a set of generators and \( R \) is a set of relations. The notation \( \langle S \mid R \rangle \) means that \( G \) is the group generated by the elements of \( S \) subject to the relations in \( R \).

Generators

The generators are elements from which every element of the group can be derived. For example, in the group of integers under addition, \( \mathbb{Z} \), the element 1 can be considered a generator because every integer can be written as a sum or difference of 1's.

Relations

The relations are equations that hold among the generators. For instance, in the cyclic group of order \( n \), denoted \( \mathbb{Z}_n \), the relation \( a^n = e \) (where \( e \) is the identity element) holds for the generator \( a \).

Examples

1. Cyclic Group: The cyclic group of order \( n \), \( \mathbb{Z}_n \), can be presented as: \[ \mathbb{Z}_n = \langle a \mid a^n = e \rangle \] 2. Free Group: The free group on two generators \( a \) and \( b \) can be presented as: \[ F_2 = \langle a, b \mid \rangle \] 3. Symmetric Group: The symmetric group on three elements, \( S_3 \), can be presented as: \[ S_3 = \langle a, b \mid a^2 = e, b^3 = e, (ab)^2 = e \rangle \]

Applications

Presentations of groups are used in various areas of mathematics, including:

• Topology: In the study of fundamental groups of topological spaces.
• Algebraic geometry: In the study of algebraic groups.
• Combinatorial group theory: In the study of groups via generators and relations.

Related Concepts

• Free group
• Cayley graph
• Group homomorphism
• Normal subgroup
• Quotient group

See Also

• Group (mathematics)
• Generator (mathematics)
• Relation (mathematics)
• Cayley graph
• Fundamental group
• Algebraic group