View source for Group action

{{TEX|done}}{{MSC|20B|22F05}}

''of a group $G$ on a set $X$''

A map from $X \times G \rightarrow X$, written $(x,g)$ or $x^g$ satisfying
$$
(x,1_G) = x
$$
$$
(x,gh) = ((x,g),h)\ .
$$
For given $g$, the map $\rho_g : x \mapsto (x,g)$ is a [[permutation]] of $X$, the [[inverse mapping]] being $\rho_{g^{-1}}$.   The map $g \mapsto \rho_g$ is a homomorphism $\rho : G \rightarrow S_X$ where $S_X$ is the [[symmetric group]] on $X$: conversely, every such homomorphism gives rise to an action $(x,g) \mapsto (x)\rho_g$.  If the homomorphism $\rho$ is injective the action is ''faithful'': $G$ may be regarded as a subgroup of $S_X$.  In any case, the image of $\rho$ is a [[permutation group]] on $X$.

If $x \in X$, the [[orbit]] of $x$ is the set of points $(x,g) : g \in G \}$.  An action is ''transitive'' if $X$ consists of a single orbit.  An action is $k$-fold transitive if for any two $k$-tuples of distinct elements $(x_1,\ldots,x_k)$ and $(y_1,\ldots,y_k)$ there is $g\in G$ such that $y_i = (x_i,g)$, $i=1,\ldots,k$.  An action is ''primitive'' if there is no non-trivial partition of $X$ preserved by $G$.  A doubly transitive action is primitive, and a primitive action is transitive, but neither converse holds.  See [[Transitive group]], [[Primitive group of permutations]].

For $x \in X$, the [[stabilizer|stabiliser]] of $x$ is the subgroup $G_x = \{ g \in G : (x,g) = x \}$.   

[[Burnside Lemma|Burnside's Lemma]] states that the number $k$ of orbits is the average number of fixed points of elements of $G$, that is, $k = |G|^{-1} \sum_{g \in G} |\mathrm{Fix}(g)|$, where $\mathrm{Fix}(g) = \{ x \in X : x^g = x \}$ and the sum is over all $g \in G$.


====References====
* P. M. Neumann, Gabrielle A. Stoy, E. C. Thompson, ''Groups and Geometry'', Oxford University Press (1994) ISBN 0-19-853451-5