# Tour:Group action

**This article adapts material from the main article:** group action

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)PREVIOUS: Subset containment gives inclusion of symmetric groups|UP: Introduction five (beginners)|NEXT: Homomorphism of groupsExpected time for this page: 20 minutes

General instructions for the tour | Pedagogical notes for the tour | Pedagogical notes for this part

WHAT YOU NEED TO DO:

- Understand the definition of group action.
- Understand the notation associated with group actions.
- Understand the meanings of orbit and stabilizer, and also convince yourself that the stabilizer of any point is a subgroup.
- Understand the example of the symmetric group acting on a set, and convince yourself that this is a group action.

## Contents

## Terminology

The term **group action** or **action of a group** is used for the notion defined here. An alternative term sometimes used is **permutation representation.**

## Definition (left action)

### Definition in terms of action

A **group action on the left** (simply called a group action if the convention is of left actions) of a group on a set is a map such that the following two conditions are satisfied:

- (here, is the identity element of ).

### Convenience of notation

For convenience, we omit the symbols or , and write the action of on as , or sometimes just as .

We can then rewrite the first condition as:

This is just like associativity, and hence we can drop the parenthesization, so we often write for either of the above.

## Terminology

### Orbit

`Further information: orbit under group action`

Suppose is a group acting on a set . Then, for any point , the orbit of under the action of , denoted , is defined as:

In other words, the orbit of a point is the set of all points that can be *reached* from that point under the action of the group.

Because of the *reversibility* of the action of elements of the group, it turns out that if is in the orbit of , is also in the orbit of . Specifically, if , then . Hence we can talk of the relation of being *in the same orbit*. This relation is reflexive (because of the identity element), symmetric (because of invertibility) and transitive (because of the homomorphism nature of the group action), and hence gives an equivalence relation. The equivalence relation thus partitions into a disjoint union of orbits.

### Transitive group action

`Further information: Transitive group action`

A group action is termed **transitive** if it has exactly one orbit (typically, we exclude the action of groups on the empty set when talking of transitive group actions, though, technically, the action on the empty set is also transitive). In other words, the action of a group on a set is termed transitive if for any , there exists such that .

### Faithful group action

`Further information: Faithful group action`

A group action is termed **faithful** if no non-identity element of the group fixes everything. In other words, the action of a group on a set is termed faithful if for every non-identity element , there exists such that .

### Stabilizer

`Further information: Point-stabilizer`

Given a group acting on a set , the **point-stabilizer** of , also termed the **isotropy group** or **isotropy subgroup** at , denoted , is defined as:

In other words, it is those elements of the group that *fix* .

## Examples

### Symmetric group action

Suppose is a set and . Then, acts on by definition: given and , we define , i.e., the image of under the permutation .

### Left-regular group action

Suppose is a group. Then, acts on *itself* by left multiplication. Here, the action is defined by:

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)PREVIOUS: Subset containment gives inclusion of symmetric groups|UP: Introduction five (beginners)|NEXT: Homomorphism of groupsExpected time for this page: 20 minutes

General instructions for the tour | Pedagogical notes for the tour | Pedagogical notes for this part