Page 82 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.

3.2 Basic notions and examples

gap> G := DihedralGroup( 8 );;
gap> ConjugacyClasses( G );
[ of ...^G, f1^G, f2^G, f3^G, f1*f2^G ]
gap> el := Elements( G );;
gap> Centralizer( G, Subgroup( G, [ el[ 5 ] ] ) );
Group([ f1*f2, f3 ])

More generally, any subgroup H ≤ G acts on G by conjugation. At the other end of the
scale, if N is a normal subgroup of G , then G , by definition, acts on N by conjugation.

Example 3.2.16 A subgroup H of a group G acts on the set of all subgroups of G by conju-
gation; (K , h) → K h . If K ≤ G , then the stabilizer of K is under this action is the normalizer
of K :

NH (K ) = {h ∈ H | K h = K }.

Example 3.2.17 Let H be a subgroup of G and H \G the set of all right cosets of H in G .
Then G acts on H \G by right multiplication: (Hx ) · g = Hx g .

gap> G := Group((1, 2, 3, 4, 5), (1, 2) );;
gap> H := Subgroup( G, [ (1, 2) ] );;
gap> Index( G, H );
60
gap> act := FactorCosetAction( G, H );

gap> Range( act );

gap> Kernel( act );
Group(())

Example 3.2.18 Let X be a non-empty set and G ≤ Sym X . Then G acts on points of X by
the rule (x , g ) → x g .

gap> G := Group( (1, 2, 3), (2, 3, 4) );;
gap> Orbit(G, 1, OnPoints);
[ 1, 2, 3, 4 ]

Let G be a finite group acting on a set X . One can observe that there is a 1-1 correspon-
dence between the elements of orbG (x ) and the right cosets of stabG (x ) in G . This implies
the following fundamental result:

Theorem 3.2.19 (Orbit-stabilizer theorem) Let G be a finite group acting on a set X .
Choose x ∈ X . Then | orbG (x )| · | stabG (x )| = |G |.

In the special case when G acts on itself by conjugation, we obtain: