Group action

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

Formally, a group action is a map from the Cartesian product G×X to X, written as $$(g,x) \mapsto gx$$ or $$xg$$ or $$x^g$$ satisfying the following properties (1G being the neutral element of G):


 * $$x^{1_G} = x ; \, $$
 * $$x^{gh} = (x^g)^h . $$

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

If we let $$A_g$$ denote the permutation associated with action by the group element $$g$$, then the map $$A : G \rightarrow S_X$$ 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


 * $$G \rightarrow G/K \rightarrow S_X, \, $$

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

Examples

 * Any group acts on any set by the trivial action in which $$x^g = x$$.
 * The symmetric group $$S_X$$ 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.

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


 * $$Stab(x) = \{ g \in G : x^g = x \} . \,$$

The stabiliser is a subgroup of G.

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


 * $$Orb(x) = \{ x^g : g \in G \} . \,$$

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


 * $$x \stackrel{G}{\sim} y \Leftrightarrow \exists g \in G, y = x^g . \, $$

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


 * $$ x^g \leftrightarrow Stab(x)g . \,$$

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 $$x^g = x$$ for all g in G: that is, such that $$Orb(x) = \{x\}$$.

Examples

 * In the trivial action, every point is a fixed point and the orbits are all singletons.
 * Let $$\pi$$ be a permutation in the usual action of $$S_n$$ on $$X = \{1,\ldots,n\}$$. The cyclic subgroup $$\langle \pi \rangle$$ generated by $$\pi$$ acts on X and the orbits are the cycles of $$\pi$$.
 * 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
An action is transitive or 1-transitive if for any x and y in X there exists a g in G such that $$y = x^g$$. 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.