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GroupTheory

  

Group

  

create a group data structure

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

Group(permutations, degreeopt, sgopt)

Group(generators, relators, embopt, sgopt)

Group(gens, operations, customopts, sgopt, lbopt)

Group(m, sgopt, embopt, lbopt)

Group(g)

Parameters

permutations

-

set, list, or sequence of permutations represented as disjoint cycles, optionally with names

degreeopt

-

(optional) equation of the form degree=n where n is a nonnegative integer; number of letters on which the permutation group acts

sgopt

-

(optional) equation of the form supergroup=g where g is a supergroup of the group to be constructed; constructs the given group as a subgroup of g

generators

-

set or list of names

relators

-

set or list of relators, each represented as a list of the generators names and their inverses

gens

-

set or list of custom group generators

operations

-

(optional) module describing group operations

customopts

-

(optional) equations of the form one = e, `1` = e, multiply = m, `.` = m, inverse = i, `/` = i, equals = eq, `=` = eq, less = o, `<` = o, or print = p

lbopt

-

(optional) equation of the form labels=lst, giving labels for the generators

m

-

Cayley table defining group operations

embopt

-

(optional) equation of the form embedding = t, where t gives the embedding

g

-

group given by permgroup or grelgroup or subgrel

Description

• 

The procedure Group creates data structures representing permutation groups, groups given by generators and relations, and Cayley table groups. The procedures PermutationGroup, FPGroup, CustomGroup, and CayleyTableGroup are specialized procedures that create only groups of the type indicated by their name. Group accepts all calling sequences for the other constructors.

• 

The first calling sequence above creates a permutation group. See PermutationGroup for more details on this calling sequence.

• 

The second calling sequence creates a finitely presented group. See FPGroup for more details on this calling sequence.

• 

The third calling sequence creates a group given by custom operations. See CustomGroup for more details on this calling sequence.

• 

The fourth calling sequence creates a group given by a Cayley table. See CayleyTableGroup for more details on this calling sequence.

• 

The final calling sequence creates either a permutation group or a finitely presented group from a group created by permgroup or grelgroup or subgrel. This is available as a backwards compatibility option.

Examples

withGroupTheory&colon;

g1Group1&comma;2&comma;1&comma;2&comma;3&comma;4&comma;5

g11&comma;2&comma;1&comma;2&comma;34&comma;5

(1)

GroupOrderg1

12

(2)

g2Groupa&equals;1&comma;2&comma;b&equals;1&comma;2&comma;3&comma;4&comma;5&comma;&apos;degree&apos;&equals;6&comma;&apos;supergroup&apos;&equals;Group1&comma;2&comma;1&comma;2&comma;3&comma;4&comma;5&comma;6

g21&comma;2&comma;1&comma;2&comma;34&comma;5

(3)

A finitely presented group and a subgroup of it, in two ways:

g3Groupa&comma;b&comma;a&comma;a&comma;a&comma;b&comma;b&comma;b&comma;a&comma;b&comma;1a&comma;1b

g3a&comma;ba3&comma;b3&comma;aba-1b-1

(4)

g4Groupc&comma;c&comma;c&comma;c&comma;&apos;supergroup&apos;&equals;g3&comma;&apos;embedding&apos;&equals;a&comma;b&comma;a

g4cc3

(5)

g5Subgroupa&comma;b&comma;a&comma;g3

g5_G_G3

(6)

The additive group of the integers modulo 6 can be constructed as a custom group in two ways: reducing to the range 0..5 after every operation or not. We present both ways here. We also show the two ways of specifying the operations: with equations and with a module.

operation_modulemoduleexport`.`a&comma;b&rarr;a&plus;bmod6&comma;`/`a&rarr;&minus;amod6&semi;end module

operation_module:=moduleexport`.`&comma;`/`&semi;end module

(7)

g6Group1&comma;operation_module

g6 < a custom group with 1 generator >

(8)

equal_modulo_6a&comma;b&rarr;iremba&comma;6&equals;0

equal_modulo_6a&comma;biremba&comma;6=0

(9)

less_modulo_6a&comma;b&rarr;amod6<bmod6

less_modulo_6a&comma;bamod6<bmod6

(10)

g7Group1&comma;`.`&equals;`+`&comma;`/`&equals;`-`&comma;`=`&equals;equal_modulo_6&comma;`<`&equals;less_modulo_6

g7 < a custom group with 1 generator >

(11)

g8Group1&verbar;2&verbar;3&verbar;4&comma;2&verbar;1&verbar;4&verbar;3&comma;3&verbar;4&verbar;1&verbar;2&comma;4&verbar;3&verbar;2&verbar;1&comma;labels&equals;a&comma;b&comma;c&comma;d

g8 < a Cayley table group with 4 elements >

(12)

g9Grouppermgroup5&comma;1&comma;2&comma;3&comma;4

g91&comma;2&comma;3&comma;4

(13)

Compatibility

• 

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

• 

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

See Also

GroupTheory

GroupTheory/references

GroupTheory[CayleyTableGroup]

GroupTheory[CustomGroup]

GroupTheory[FPGroup]

GroupTheory[PermutationGroup]

GroupTheory[Subgroup]