36.6 Cosets

returns the right coset of U with representative g, which is the set of all elements of the form ug for all u Î U . g must be an element of a larger group G which contains U. For element operations such as in a right coset behaves like a set of group elements.

Right cosets are external orbits for the action of U which acts via OnLeftInverse. Of course the action of a larger group G on right cosets is via OnRight.

gap> u:=Group((1,2,3), (1,2));;c:=RightCoset(u,(2,3,4));
RightCoset(Group( [ (1,2,3), (1,2) ] ),(2,3,4))
gap> ActingDomain(c);
Group( [ (1,2,3), (1,2) ] )
gap> Representative(c);
(2,3,4)
gap> Size(c);
6
gap> AsList(c);
[ (2,3,4), (1,4,2), (1,3,4,2), (1,3)(2,4), (2,4), (1,4,2,3) ]

computes a duplicate free list of right cosets Ug for g Î G . A set of representatives for the elements in this list forms a right transversal of U in G. (By inverting the representatives one obtains a list of representatives of the left cosets of U.) The NC version does not check whether U is a subgroup of G.

gap> RightCosets(g,u);
[ RightCoset(Group( [ (1,2,3), (1,2) ] ),()), 
  RightCoset(Group( [ (1,2,3), (1,2) ] ),(1,2)(3,4)), 
  RightCoset(Group( [ (1,2,3), (1,2) ] ),(1,3)(2,4)), 
  RightCoset(Group( [ (1,2,3), (1,2) ] ),(1,4)(2,3)) ]

returns a ``canonical'' representative of the coset Ug which is independent of the given representative g. This can be used to compare cosets by comparing their canonical representatives. The representative chosen to be the ``canonical'' one is representation dependent and only guaranteed to remain the same within one GAP session.

gap> CanonicalRightCosetElement(u,(2,4,3));
(3,4)

The category of right cosets.

GAP does not provide left cosets as a separate data type, but as the left coset gU consists of exactly the inverses of the elements of the right coset Ug^-1 calculations with left cosets can be emulated using right cosets by inverting the representatives.

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