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.
GAP 3.4.4