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