# The 30 Subgroups of the Symmetric Group on Four Symbols

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

The symmetric group on a finite set of symbols is the group whose elements are all permutations of the symbols. The group operation is the composition of such permutations, which are bijective functions from the set of symbols to itself.

Contributed by: Gerard Balmens (January 2014)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

A group is a set together with an operation (called the group law of ) that combines any two elements and to form another element, denoted or . To qualify as a group, the set and operation must satisfy four requirements known as the group axioms: closure, associativity, identity element, and inverse element. If for all and in , then the group is commutative (or abelian).

In this Demonstration, the group law is the composition of permutations of the set . For example, .

## Permanent Citation