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 operations on the set.

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

$x^{1_{G}}=x;\,$

$x^{gh}=(x^{g})^{h}.$

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 (group theory)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 distict 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.