Symmetric group

Written by Patrick Stevens last updated
Requires: Function

Summaries

The notion that group theory captures the idea of "symmetry" derives from the notion of the symmetric group, and the very important theorem due to Cayley that every group is a subgroup of a symmetric group.

Definition

Let be a set. A bijection is a permutation of . Write for the set of permutations of the set (so its elements are functions).

Then is a group under the operation of composition of functions; it is the symmetric group on . (It is also written , for the automorphism group.)

We write for , the symmetric group on elements.

Elements of

We can represent a permutation of in two different ways, each of which is useful in different situations.

Double-row notation

Let , so is a function . Then we write for . This has the advantage that it is immediately clear where every element goes, but the disadvantage that it is quite hard to see the properties of an element when it is written in double-row notation (for example, " cycles round five elements" is hard to spot at a glance), and it is not very compact.

Cycle notation

Cycle notation is a different notation, which has the advantage that it is easy to determine an element's order and to get a general sense of what the element does. Every element of can be expressed in (disjoint) cycle notation in an essentially unique way.

Product of transpositions

It is a useful fact that every permutation in a (finite) symmetric group may be expressed as a product of transpositions.

Examples

  • The group is the group of permutations of a one-point set. It contains the identity only, so is the trivial group.
  • The group is isomorphic to the cyclic group of order . It contains the identity map and the map which interchanges and .

Those are the only two abelian symmetric groups. Indeed, in cycle notation, and do not commute in for , because while .

  • The group contains the following six elements: the identity, . It is isomorphic to the dihedral group on three vertices. (Proof.)

Why we care about the symmetric groups

A very important (and rather basic) result is Cayley's Theorem, which states the link between group theory and symmetry.

Relationship to the alternating group

The alternating group is defined as the collection of elements of which can be made by an even number of transpositions. This does form a group (proof).