The 30 Subgroups of the Symmetric Group on Four Symbols

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.

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, .