factor a group element into a subgroup element and a coset representative
Factor( g, H )
permutation or word on the generators of the supergroup of H
a permutation group or a subgroup of a finitely presented group
Let H be a subgroup of a group G, and let g be a member of G. Let R be a complete set of representatives of the right cosets of H in G. Then g can be written, uniquely, in the form g=h·r, with h in H and r in R.
The Factor( g, H ) command returns a pair [ h, r ], where h belongs to H, and r is a coset representative for the coset H.g in a supergroup of H.
If H is a permutation group, then the representative is for the cosets of H in the full symmetric group of the same degree as H. If H is a subgroup of a finitely presented group G, then the representative r is for the cosets of H in G.
The set of representatives used is the set obtained from the RightCosets command applied to H.
First we consider the following subgroup of the symmetric group of degree 7.
H ≔ Group⁡Perm⁡1,2,3,Perm⁡3,4,5,6,7
We can factor this permutation over the cosets of H in Symm(7).
g ≔ Perm⁡3,4,5,6
f ≔ Factor⁡g,H
R ≔ map⁡Representative,RightCosets⁡H,Symm⁡7
Next, consider the group of the (2,3)-torus knot, which is an infinite group.
G ≔ a,b|a2=b3
The following subgroup of G has index in G equal to 3.
H ≔ Subgroup⁡a,`.`⁡b−1,a,b,`.`⁡b,a,b−1,G
The alternating group of degree 5 has the following presentation.
G ≔ a,b|a2,b3,`.`⁡a,b5=1
H ≔ Subgroup⁡`.`⁡a,b,G
The GroupTheory[Factor] command was introduced in Maple 17.
For more information on Maple 17 changes, see Updates in Maple 17.
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