Stabilizer( G, d )
Stabilizer( G, d, operation )
Stabilizer returns the stabilizer of the point d under the operation
of the group G.
The stabilizer S of d in G is the subgroup of those elements g
of G that fix d, i.e., for which d^g = d. The right cosets of S
correspond in a canonical way to the points p in the orbit O of d
under G; namely all elements from a right coset S g map d to the
same point d^g in O, and elements from different right cosets S g
and S h map d to different points d^g and d^h. Thus the index of
the stabilizer S in G is equal to the length of the orbit O.
RepresentativesOperation (see RepresentativesOperation) computes a
system of representatives of the right cosets of S in G.
Stabilizer accepts a function operation of two arguments d and g
as optional third argument, which specifies how the elements of G
operate (see Other Operations).
gap> g := Group( (1,2,3)(6,7), (3,4,5)(7,8) );;
gap> g.name := "G";;
gap> Stabilizer( g, 1 );
Subgroup( G, [ (3,4,5)(7,8), (2,5,3)(6,7) ] )
gap> Stabilizer( g, [1,2,3], OnSets );
Subgroup( G, [ (7,8), (6,8), (2,3)(4,5)(6,7,8), (1,2)(4,5)(6,7,8) ] )
Stabilizer calls
G.operations.Stabilizer( G, d, operation )
and returns the value. Note that the third argument is not optional for
functions called this way.
The default function called this way is GroupOps.Stabilizer, which
computes the orbit of d under G, remembers a representative r_e for
each point e in the orbit, and uses Schreier's theorem, which says
that the stabilizer is generated by the elements r_e g r_{e^g}^{-1}.
Special categories of groups overlay this default function with more
efficient functions.
GAP 3.4.4