Group action

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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 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 $x g$ or $x g$ satisfying the following properties (1_G being the neutral element of G):

$x^{1_G} = x ; \,$

x g h = (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.

[edit] 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.

[edit] 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.

[edit] 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 $O r b (x) = {x}$ .

[edit] Examples

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 $S n$ on $X = \{1,\ldots,n\}$ . The cyclic subgroup $\langle \pi \rangle$ 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.

[edit] 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.

Some content on this page may previously have appeared on Citizendium.

Group action

Contents

[edit] Examples

[edit] Stabilisers

[edit] Orbits

[edit] Examples

[edit] Transitivity

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