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GroupTheory

  

SearchSmallGroups

  

search for small groups satisfying specified properties

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

SearchSmallGroups(spec, formopt)

Parameters

spec

-

expression sequence of search parameters

formopt

-

(optional) an option of the form form = X, where X is one of "id" (the default), "permgroup", "fpgroup" or "count".

Description

• 

The SearchSmallGroups(spec) command searches Maple's small groups database for groups satisfying properties specified in a sequence spec of search parameters. The valid search parameters may be grouped into several classes, as follows.

• 

Use the form = X option to control the form of the output from this command. By default, an expression sequence of IDs for the Small groups database is returned. This is the same as specifying form = "id". To have an expression sequence of groups, either permutation groups, or finitely presented groups, use either the form = "permgroup" or form = "fpgroup" options, respectively. Finally, the form = "count" option causes SearchSmallGroups to return just the number of groups in the database satisfying the constraints implied by the search parameters.

• 

Note that the IDs returned in the default case are the IDs of the groups within the SmallGroups database.  These may differ from the IDs for the same group if it happens to be present in another database, which has its own set of group IDs.

Boolean Search Parameters

• 

Boolean search parameters p, such as abelian, can be specified in one of the forms p = true, p = false, or just p (which is equivalent to p = true). If the boolean search parameter p is true, then only groups satisfying the corresponding predicate are returned.  If the boolean search parameter p is false, then only groups that do not satisfy the predicate are returned. Leaving a boolean search parameter unspecified causes the SearchSmallGroups command to return groups that do, and do not, satisfy the corresponding predicate.

• 

The supported boolean search parameters are described in the following table.

cyclic

describes the class of cyclic groups

abelian

describes the class of Abelian groups

elementary

describes the class of elementary Abelian groups

homocyclic

describes the class of homocyclic groups

nilpotent

describes the class of nilpotent groups

hallpaige

describes the class of Hall-Paige groups

hamiltonian

describes the class of Hamiltonian groups

metacyclic

describes the class of metacyclic groups

metabelian

describes the class of metabelian groups

lagrangian

describes the class of Lagrangian (CLT) groups

clt

describes the class of Lagrangian (CLT) groups

gclt

describes the class of GCLT groups

supersoluble

describes the class of supersoluble groups

supersolvable

describes the class of supersoluble groups

sylowtower

describes the class of groups having a Sylow tower

orderedsylowtower

describes the class of groups having an ordered Sylow tower

soluble

describes the class of soluble groups

perfect

describes the class of perfect groups

complete

describes the class of complete groups

simple

describes the class of simple groups

almostsimple

describes the class of almost simple groups

indecomposable

describes the class of directly indecomposable groups

Numeric Search Parameters

• 

Maple supports several search parameters that describe numeric invariants of finite groups. All have positive integral values. A numeric search parameter p may be given in the form p = n, for some specific value n, or by indicating a range, as in p = a .. b. In the former case, only groups for which the numeric parameter has the value n will be returned. In the case in which a range is specified, groups for which the numeric invariant lies within the indicate range (inclusive of its end-points) are returned.

• 

The supported numeric search parameters are listed in the following table.

order

indicates the order (cardinality) of the group

exponent

indicates the exponent of the group

nilpotencyclass

indicates the class of a nilpotent group

derivedlength

indicates the length of the derived series of a soluble group

fittinglength

indicates the length of the lower Fitting series of a soluble group

frattinilength

indicates the length of the Frattini series of a group

compositionlength

indicates the length of a composition series of a group

chieflength

indicates the length of a chief series of a group

classnumber

indicates the number of conjugacy classes of the group

orderclassnumber

indicates the number of order classes of the group

elementordersum

indicates the sum of the orders of the group elements

maxelementorder

indicates the largest order of a group element

nsylow[p]

indicates the number of Sylow p-subgroups

Subgroup and Quotient Group Search Parameters

• 

Several subgroup search parameters are supported. These describe the isomorphism type of various subgroups of a group by specifying the Small Group ID (as returned by the IdentifySmallGroup command).

• 

For a subgroup or quotient search parameter p, passing an equation of the form p = [ord,id] causes the SearchSmallGroups command to return only groups whose subgroup, or quotient group, corresponding to p are isomorphic to the small group ord/id to be returned.  Passing an equation of the form p = ord causes the SearchSmallGroups command to return only groups whose subgroup, or quotient group, corresponding to p have order ord.

• 

The following table describes the supported subgroup and quotient group search parameters.

center

specifies the center of a group

derivedsubgroup

specifies the derived, or commutator, subgroup of a group

fittingsubgroup

specifies the Fitting subgroup of a group

frattinisubgroup

specifies the Frattini subgroup of a group

socle

specifies the socle of a group

sylow[p]

specifies the Sylow p-subgroup of a group, where p is a prime number

derivedquotient

specifies the derived quotient G/[G,G]

centralquotient

specifies the central quotient G/Z(G)

frattiniquotient

specifies the Frattini quotient G/Fratt(G)

fittingquotient

specifies the Fitting quotient G/Fitt(G)

soclequotient

specifies the socle quotient G/Socle(G)

• 

Note that the sylow[p] option is always used with a prime number index p.

Examples

withGroupTheory:

Count the total number of groups in the database.

SearchSmallGroupsform=count

92804

(1)

Find all the nilpotent groups of order 24.

SearchSmallGroupsorder=24,nilpotent

24,2,24,9,24,10,24,11,24,15

(2)

Find all the non-Abelian groups of order 10, as permutation groups.

SearchSmallGroupsorder=10,abelian=false,form=permgroup

1,23,64,57,108,9,1,3,7,8,42,5,9,10,6

(3)

Find those finite groups in the database whose Sylow 2-subgroup is the cyclic group of order 4, and whose Sylow 3-subgroup is the small group 27/4.

SearchSmallGroupssylow2=4,1,sylow3=27,4

108,9,108,14

(4)

Find the perfect groups of order at most 100.

SearchSmallGroupsorder=1..100,perfect

1,1,60,5

(5)

Verify that there is no non-Abelian simple group with order less than 60.

SearchSmallGroupssimple&comma;abelian=false&comma;order<60

Find all the non-Abelian simple groups in the database.

SearchSmallGroupssimple&comma;abelian=false

60&comma;5,168&comma;42,360&comma;118,504&comma;156

(6)

Find the groups up to order 24 whose Sylow 2-subgroup has order 4.

SearchSmallGroupsorder=1..24&comma;sylow2=4

4&comma;1,4&comma;2,12&comma;1,12&comma;2,12&comma;3,12&comma;4,12&comma;5,20&comma;1,20&comma;2,20&comma;3,20&comma;4,20&comma;5

(7)

Find the perfect groups in the database that are not simple.

SearchSmallGroupsperfect&comma;simple=false

1&comma;1,120&comma;5,336&comma;114

(8)

Find the non-Abelian groups of order 8.

SearchSmallGroupsorder=8&comma;abelian=false

8&comma;3,8&comma;4

(9)

Show that neither of these occurs as the Frattini subgroup of any group in the database. (In fact, neither occurs as the Frattini subgroup of any finite group.)

SearchSmallGroupsfrattinisubgroup=8&comma;3

SearchSmallGroupsfrattinisubgroup=8&comma;4

LSearchSmallGroupsmaxelementorder=5&colon;

mapirem&comma;map2op&comma;1&comma;L&comma;5

0&comma;0&comma;0&comma;0&comma;0&comma;0&comma;0&comma;0&comma;0&comma;0&comma;0&comma;0&comma;0&comma;0&comma;0&comma;0&comma;0&comma;0&comma;0

(10)

seqSearchSmallGroupsmaxelementorder=i&comma;form=count&comma;i=2..10

8,18,31413,19,361,21,24625,182,149

(11)

Find the groups of order at most 32 whose Sylow 2-subgroup and Sylow 3-subgroup is normal.

SearchSmallGroupsorder=1..32&comma;nsylow2=1&comma;nsylow3=1

6&comma;2,12&comma;2,12&comma;5,18&comma;2,18&comma;5,24&comma;2,24&comma;9,24&comma;10,24&comma;11,24&comma;15,30&comma;4

(12)

Find the perfect, but non-simple groups in the database whose central quotient has order 168.

SearchSmallGroupssimple=false&comma;perfect&comma;centralquotient=168

336&comma;114

(13)

The form = "count" option is particularly handy for developing visualizations depicting the relative frequency of groups satisfying particular properties.

Statistics:-PieChartseqi=SearchSmallGroupsfittinglength=i&comma;form=count&comma;i=1..3

plots:-displayStatistics:-PieChartseqi=SearchSmallGroupsderivedlength=i&comma;form=count&comma;i=1..5&comma;annular=1&comma;Statistics:-PieChartseqi=SearchSmallGroupsnilpotencyclass=i&comma;form=count&comma;i=1..7&comma;annular=2&comma;Statistics:-PieChartseqi=SearchSmallGroupscompositionlength=i&comma;form=count&comma;i=1..8&comma;annular=3

Compatibility

• 

The GroupTheory[SearchSmallGroups] command was introduced in Maple 2015.

• 

For more information on Maple 2015 changes, see Updates in Maple 2015.

• 

The spec parameter was updated in Maple 2019.

• 

The form option was updated in Maple 2019.

• 

The GroupTheory[SearchSmallGroups] command was updated in Maple 2020.

See Also

GroupTheory

GroupTheory[Center]

GroupTheory[DerivedLength]

GroupTheory[DerivedSubgroup]

GroupTheory[Exponent]

GroupTheory[FittingSubgroup]

GroupTheory[FrattiniSubgroup]

GroupTheory[GroupOrder]

GroupTheory[IdentifySmallGroup]

GroupTheory[IsAbelian]

GroupTheory[IsNilpotent]

GroupTheory[IsPerfect]

GroupTheory[IsSimple]

GroupTheory[IsSoluble]

GroupTheory[NilpotencyClass]

GroupTheory[SmallGroup]