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GroupTheory

  

LeftCoset

  

construct the left coset of a subgroup

  

RightCoset

  

construct the right coset of a subgroup

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

LeftCoset( g, H )

RightCoset( H, g )

Parameters

g

-

an element of some group containing H

H

-

a subgroup of a permutation group

Description

• 

Let H be a subgroup of a group G, and let g be a member of G. The left coset gH is defined to be the subset gh:hH of G. Similarly, the right coset Hg is the subset hg:hH of G.

• 

The LeftCoset( g, H ) command returns a data structure representing the left coset gH of a subgroup H of a permutation group G. The RightCoset( H, g ) command returns a data structure representing the right coset Hg of a subgroup H of a permutation group G.

• 

The data structures representing (left or right) cosets respond to the following methods.

Representative( c )

returns the representative of the coset c

numelems( c )

returns the number of members of the coset c

member( x, c ) or x in c

returns true if x belongs to the coset c

Elements( c )

returns the elements of the coset c, as a set

Subgroup( c )

returns the subgroup of the coset c

Examples

withGroupTheory:

GAlt4

GA4

(1)

HSubgroup1,2,3,4,G

H1,23,4

(2)

CRightCosetH,1,2,3

C1,23,4·1,2,3

(3)

GroupOrderH

2

(4)

numelemsC

2

(5)

ElementsC

1,2,3,1,3,4

(6)

The symmetric group on 3 letters.

M1|2|3|4|5|6,2|1|6|5|4|3,3|5|1|6|2|4,4|6|5|1|3|2,5|3|4|2|6|1,6|4|2|3|1|5

M123456216543351624465132534261642315

(7)

GCayleyTableGroupM

G < a Cayley table group with 6 elements >

(8)

2 is an involution.

HSubgroup2&comma;G

H < a Cayley table group with 1 generator >

(9)

RCRightCosetH&comma;3

RC < a Cayley table group with 1 generator > ·3

(10)

numelemsH=numelemsRC

2=2

(11)

ElementsRC

3&comma;6

(12)

LCLeftCoset3&comma;H

LC3· < a Cayley table group with 2 elements >

(13)

numelemsH=numelemsLC

2=2

(14)

ElementsLC

3&comma;5

(15)

Compatibility

• 

The GroupTheory[LeftCoset] and GroupTheory[RightCoset] commands were introduced in Maple 17.

• 

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

See Also

GroupTheory

GroupTheory[Cosets]