Group action

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This editable Main Article is under development and subject to a disclaimer.

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In mathematics, a group action is a relation between a group G and a set X in which the elements of G act as operations on the set.

Formally, a group action is a map from the Cartesian product G × X -> X {\displaystyle G\times X\rightarrow X} , written as  ( g , x ) ↦ g x {\displaystyle (g,x)\mapsto gx} or  x g {\displaystyle xg} or  x g {\displaystyle x^{g}} satisfying the following properties:

x 1 G = x ; {\displaystyle x^{1_{G}}=x;\,}
x g h = ( x g ) h . {\displaystyle x^{gh}=(x^{g})^{h}.}

From these we deduce that  ( x g − 1 ) g = x g − 1 g = x 1 G = x {\displaystyle \left(x^{g^{-1}}\right)^{g}=x^{g^{-1}g}=x^{1_{G}}=x} , so that each group element acts as an invertible function on X, that is, as a permutation of X.

If we let  A g {\displaystyle A_{g}} denote the permutation associated with action by the group element  g {\displaystyle g} , then the map  A : G -> S X {\displaystyle A:G\rightarrow S_{X}} from G to the symmetric group on X is a group homomorphism, and every group action arises in this way. We may speak of the action as a permutation representation of G. The kernel of the map A is also called the kernel of the action, and a faithful action is one with trivial kernel. Since we have

G -> G / K -> S X , {\displaystyle G\rightarrow G/K\rightarrow S_{X},\,}

where K is the kernel of the action, there is no loss of generality in restricting consideration to faithful actions where convenient.

## Contents

* 1 Examples
* 2 Stabilisers
* 3 Orbits
* 3.1 Examples
* 4 Transitivity

## Examples[edit]

* Any group acts on any set by the trivial action in which  x g = x {\displaystyle x^{g}=x} .
* The symmetric group  S X {\displaystyle S_{X}} acts of X by permuting elements in the natural way.
* The automorphism group of an algebraic structure acts on the structure.
* A group acts on itself by right translation.
* A group acts on itself by Conjugation (group theory)conjugation.

## Stabilisers[edit]

The stabiliser of an element x of X is the subset of G which fixes x:

S t a b ( x ) = { g ∈ G : x g = x } . {\displaystyle Stab(x)=\\{g\in G:x^{g}=x\\}.\,}

The stabiliser is a subgroup of G.

## Orbits[edit]

The orbit of any x in X is the subset of X which can be "reached" from x by the action of G:

O r b ( x ) = { x g : g ∈ G } . {\displaystyle Orb(x)=\\{x^{g}:g\in G\\}.\,}

The orbits partition the set X: they are the equivalence classes for the relation  ∼ G {\displaystyle {\stackrel {G}{\sim }}} defined by

x ∼ G y ⇔ ∃ g ∈ G , y = x g . {\displaystyle x{\stackrel {G}{\sim }}y\Leftrightarrow \exists g\in G,y=x^{g}.\,}

If x and y are in the same orbit, their stabilisers are conjugate.

The elements of the orbit of x are in one-to-one correspondence with the right cosets of the stabiliser of x by

x g ↔ S t a b ( x ) g . {\displaystyle x^{g}\leftrightarrow Stab(x)g.\,}

Hence the order of the orbit is equal to the index of the stabiliser. If G is finite, the order of the orbit is a factor of the order of G.

A fixed point of an action is just an element x of X such that  x g = x {\displaystyle x^{g}=x} for all g in G: that is, such that  O r b ( x ) = { x } {\displaystyle Orb(x)=\\{x\\}} .

### Examples[edit]

* In the trivial action, every point is a fixed point and the orbits are all singletons.
* Let  π {\displaystyle \pi } be a permutation in the usual action of  S n {\displaystyle S_{n}} on  X = { 1 , … , n } {\displaystyle X=\\{1,\ldots ,n\\}} . The cyclic subgroup  ⟨ π ⟩ {\displaystyle \langle \pi \rangle } generated by  π {\displaystyle \pi } acts on X and the orbits are the cycles of  π {\displaystyle \pi } .
* If G acts on itself by conjugation, then the orbits are the conjugacy classes and the fixed points are the elements of the centre.

## Transitivity[edit]

An action is transitive or 1-transitive if for any x and y in X there exists a g in G such that  y = x g {\displaystyle y=x^{g}} . Equivalently, the action is transitive if it has only one orbit.

More generally an action is k -transitive for some fixed number k if any k-tuple of distinct elements of X can be mapped to any other k-tuple of distinct elements by some group element.

An action is primitive if there is no non-trivial partition of the set X which is preserved by the group action. Since the orbits form a partition preserved by this group action, primitive implies transitive. Further, 2-transitive implies primitive.