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GroupTheory

  

Factor

  

factor a group element into a subgroup element and a coset representative

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

Factor( g, H )

Parameters

g

-

permutation or word on the generators of the supergroup of H

H

-

a permutation group or a subgroup of a finitely presented group

Description

• 

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.

Examples

withGroupTheory:

First we consider the following subgroup of the symmetric group of degree 7.

HGroupPerm1,2,3,Perm3,4,5,6,7

H1,2,3,3,4,5,6,7

(1)

We can factor this permutation over the cosets of H in Symm(7).

gPerm3,4,5,6

g3,4,5,6

(2)

fFactorg,H

f3,4,5,7,6,6,7

(3)

RmapRepresentative,RightCosetsH,Symm7

R,6,7

(4)

memberf2,R

true

(5)

f1·f2

3,4,5,6

(6)

Next, consider the group of the (2,3)-torus knot, which is an infinite group.

Ga,b|a2=b3

Ga,ba-2b3

(7)

The following subgroup of G has index in G equal to 3.

HSubgroupa,b·a·b1,b1·a·b,G

H_G,_G0,_G1_G2_G0-2,_G1_G0-2_G1_G0-2_G2

(8)

Factorb·a·b,H

a2ba-1b-1a2,b-1

(9)

Factora·b,H

a3ba-2b-1,b

(10)

Factorb·a22,H

a2ba-1b-1a2ba-1b-1a2b-1a2b,b-1

(11)

The alternating group of degree 5 has the following presentation.

Ga,b|a2,b3,a·b5=1

Ga,ba2,b3,ababababab

(12)

HSubgroupa·b,G

H_G2_G25

(13)

Factora·b·b,H

ab,b

(14)

Compatibility

• 

The GroupTheory[Factor] command was introduced in Maple 17.

• 

For more information on Maple 17 changes, see Updates in Maple 17.

See Also

GroupTheory

GroupTheory[Group]

GroupTheory[RightCosets]

GroupTheory[SymmetricGroup]