Group action

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

[edit intro]

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, written as (g,x)↦gx or xg or xg satisfying the following properties:

x1G=x;
xgh=(xg)h.

From these we deduce that (xg−1)g=xg−1g=x1G=x, so that each group element acts as an invertible function on X, that is, as a permutation of X.

If we let Ag denote the permutation associated with action by the group element g, then the map A:G->SX 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->SX,

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 xg=x.
* The symmetric group SX 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:

Stab(x)={g∈G:xg=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:

Orb(x)={xg:g∈G}.

The orbits partition the set X: they are the equivalence classes for the relation ∼G defined by

x∼Gy⇔∃g∈G,y=xg.

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

xg↔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 xg=x for all g in G: that is, such that Orb(x)={x}.

### Examples[edit]

* In the trivial action, every point is a fixed point and the orbits are all singletons.
* Let π be a permutation in the usual action of Sn on X={1,…,n}. The cyclic subgroup ⟨π⟩ generated by π acts on X and the orbits are the cycles of π.
* 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=xg. 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.