Symmetric group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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To get more information on particular symmetric groups, check out Category:Symmetric groups

Definition

Symbol-free definition

The symmetric group on a set is the group of all permutations of that set (i.e., bijective maps from the set to itself). A group is said to be a symmetric group if it is isomorphic to the symmetric group on some set.

Definition with symbols

The symmetric group over a set $S$ (denoted as $Sym(S)$) is defined as the group of all permutations on $S$, with the multiplication being function composition.

A group $G$ is termed a symmetric group if $G\cong Sym(S)$ for some set $S$.

A bijection between sets gives rise to an isomorphism of the corresponding symmetric groups. Thus, there is only one symmetric group, upto isomorphism, on a set of given cardinality. The symmetric group on a set of cardinality $n$ is denoted $Sym(n)$, or sometimes $S_{n}$.

Relation with other properties

Weaker properties

• Centerless group: All except the symmetric group on two elements are centerless. Further information: Symmetric groups are centerless
• Complete group: All except the symmetric group on two elements and the symmetric group on six elements. Further information: Symmetric groups are complete

IAPS structure

Further information: Permutation IAPS