# 7.116 Operations for Groups

G ^ s

The operator ^ evaluates to the subgroup conjugate to G under a group element s of the parent group of G. See ConjugateSubgroup.

gap> s4 := Group( (1,2,3,4), (1,2) );
Group( (1,2,3,4), (1,2) )
gap> s4.name := "s4";;
gap> v4 := Subgroup( s4, [ (1,2), (1,2)(3,4) ] );
Subgroup( s4, [ (1,2), (1,2)(3,4) ] )
gap> v4 ^ (2,3);
Subgroup( s4, [ (1,3), (1,3)(2,4) ] )
gap> v4 ^ (2,5);
Error, <g> must be an element of the parent group of <G>

s in G

The operator in evaluates to true if s is an element of G and false otherwise. s must be an element of the parent group of G.

gap> (1,2,3,4) in v4;
false
gap> (2,4) in v4^(2,3);
true

G * s

The operator * evaluates to the right coset of G with representative s. s must be an element of the parent group of G. See RightCoset for details about right cosets.

s * G

The operator * evaluates to the left coset of G with representative s. s must be an element of the parent group of G. See LeftCoset for details about left cosets.

gap> v4 * (1,2,3,4);
(Subgroup( s4, [ (1,2), (1,2)(3,4) ] )*(1,2,3))
gap> (1,2,3,4) * v4;
((1,2,3,4)*Subgroup( s4, [ (1,2), (1,2)(3,4) ] ))

G / N

The operator / evaluates to the factor group <G> / <N> where N must be a normal subgroup of G. This is the same as FactorGroup(G,N) (see FactorGroup).

GAP 3.4.4
April 1997