7.89 LeftCosets

LeftCosets( G, U )

LeftCosets returns a list of the left cosets of the subgroup U in the group G. The list is not sorted, i.e., the left cosets may appear in any order. The left cosets are domains as constructed by LeftCosets (see LeftCosets).

    gap> G := Group( (1,2), (1,2,3,4) );;
    gap> G.name := "G";;
    gap> U := Subgroup( G, [ (1,2), (3,4) ] );;
    gap> LeftCosets( G, U );
    [ (()*Subgroup( G, [ (1,2), (3,4) ] )),
      ((2,3,4)*Subgroup( G, [ (1,2), (3,4) ] )),
      ((2,3)*Subgroup( G, [ (1,2), (3,4) ] )),
      ((1,3,4,2)*Subgroup( G, [ (1,2), (3,4) ] )),
      ((1,3,2)*Subgroup( G, [ (1,2), (3,4) ] )),
      ((1,3)(2,4)*Subgroup( G, [ (1,2), (3,4) ] )) ] 

If G is the parent of U, the dispatcher LeftCosets first checks whether U has a component leftCosets. If U has this component, it returns that value. Otherwise LeftCosets calls G.operations.LeftCosets(G,U), remembers the returned value in U.leftCosets and returns it. If G is not the parent of U, LeftCosets calls G.operations.LeftCosets(G,U) directly and returns that value.

The default function called this way is GroupOps.LeftCosets, which calls RightCosets( G, U ) and turns each right coset U * u into the left coset u^-1 * U. Look up the entries for LeftCosets in the index, to see for which groups this function is overlaid.

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