create a group data structure
Group(permutations, degreeopt, sgopt)
Group(generators, relators, embopt, sgopt)
Group(gens, operations, customopts, sgopt, lbopt)
Group(m, sgopt, embopt, lbopt)
set, list, or sequence of permutations represented as disjoint cycles, optionally with names
(optional) equation of the form degree=n where n is a nonnegative integer; number of letters on which the permutation group acts
(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
set or list of names
set or list of relators, each represented as a list of the generators names and their inverses
set or list of custom group generators
(optional) module describing group operations
(optional) equations of the form one = e, `1` = e, multiply = m, `.` = m, inverse = i, `/` = i, equals = eq, `=` = eq, less = o, `<` = o, or print = p
(optional) equation of the form labels=lst, giving labels for the generators
Cayley table defining group operations
(optional) equation of the form embedding = t, where t gives the embedding
group given by permgroup or grelgroup or subgrel
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.
g1 ≔ Group⁡1,2,1,2,3,4,5
g2 ≔ Group⁡a=1,2,b=1,2,3,4,5,'degree'=6,'supergroup'=Group⁡1,2,1,2,3,4,5,6
A finitely presented group and a subgroup of it, in two ways:
g3 ≔ Group⁡a,b,a,a,a,b,b,b,a,b,1a,1b
g4 ≔ Group⁡c,c,c,c,'supergroup'=g3,'embedding'=a,b,a
g5 ≔ Subgroup⁡a,b,a,g3
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_module ≔ moduleexport`.` ≔ a,b→a+bmod6,`/` ≔ a→−amod6;end module
g6 ≔ Group⁡1,operation_module
g6≔ < a custom group with 1 generator >
equal_modulo_6 ≔ a,b→irem⁡b−a,6=0
less_modulo_6 ≔ a,b→amod6<bmod6
g7 ≔ Group⁡1,`.`=`+`,`/`=`-`,`=`=equal_modulo_6,`<`=less_modulo_6
g7≔ < a custom group with 1 generator >
g8 ≔ Group⁡1|2|3|4,2|1|4|3,3|4|1|2,4|3|2|1,labels=a,b,c,d
g8≔ < a Cayley table group with 4 elements >
g9 ≔ Group⁡permgroup⁡5,1,2,3,4
The GroupTheory[Group] command was introduced in Maple 17.
For more information on Maple 17 changes, see Updates in Maple 17.
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