Group action

2020 Mathematics Subject Classification: Primary: 20B Secondary: 22F05 [MSN][ZBL]

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 stabiliser of $x$ is the subgroup $G_x = \{ g \in G : (x,g) = x \}$.

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