Group Actions

(iii) Let V be a vector space of dimension n over a field F . Any lineartransformation T : V → V can be represented by an n × n matrixwith entries from F and the transformation is invertible when thecorresponding matrix is non-singular (i.e., has non-zero determinant).Recall that the general linear group of degree n over F isGL n (F )={ A | A is an n × n matrix over F with det A ≠0}.Then GL n (F )actsonV : a matrix A in GL n (F )movesthevectorv(from V )accordingtothelineartransformationdeterminedbyA.These give us examples of group actions where Ω is a combinatorialobject (a set), a geometric object (an n-gon) and a vector space. We willalso consider lots of examples of groups acting on something related to theirown structure, such as their subgroups or their elements. Let’s first developthe basic theory of group actions, and then consider applications.2.2 OrbitsDefinition 2.3 Let the group G act on the set Ω. The orbit of ω ∈ Ωisω G = { ω x | x ∈ G }⊆Ω.Thus, the orbit of ω consists of all the points of Ω we can get to byapplying elements of the group G to ω.Example 2.4 Consider the dihedral group D 2n from Example 2.2, actingon the vertices of the n-gon. The permutation α rotates the verticesanticlockwise by one vertex. Consider vertex number 1. By actingwith α, wecaneventuallymovethisvertextoeveryothervertex. Hence1 D 2n= {1,...,n}.The basic properties of orbits are as follows.Proposition 2.5 Let the group G act on the set Ω, andletα, β ∈ Ω. Then(i) α ∈ α G ;(ii) either α G = β G or α G ∩ β G = ∅.Thus part (ii) asserts that any two orbits are either disjoint orareequal.The proposition then yields:Corollary 2.6 Let the group G act on the set Ω. Then Ω is the disjointunion of the orbits of G.□23