GroupTheory - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Group Theory : GroupTheory/IdentifySmallGroup

GroupTheory

  

IdentifySmallGroup

  

find where a group is in the small groups database

 

Calling Sequence

Parameters

Options

Description

Examples

Compatibility

Calling Sequence

IdentifySmallGroup(G, opts)

Parameters

G

-

a group

opts

-

(optional) equations of the form keyword = value, listed below

Options

• 

assign = name

  

If given the option assign = x, where x is any name, IdentifySmallGroup will assign the isomorphism mapping G to H to the name x. This isomorphism can be used in the same way as the isomorphisms assigned by AreIsomorphic.

  

If x already has a value, then it needs to be protected from evaluation using quotation marks.

• 

form = fpgroup or form = permgroup

  

This option can be used together with the assign option, explained above, in order to specify the form of the group H that is the codomain of the isomorphism to be assigned to the name specified in the assign option.

  

Specifying form = fpgroup results in the codomain being a finitely presented group. Specifying form = permgroup (the default) results in the codomain being a permutation group. You can equivalently specify the string forms of these values, as form = "fpgroup" or form = "permgroup".

  

If no assign option is specified, then the form option is ignored.

Description

• 

The command IdentifySmallGroup finds if a group H isomorphic to G occurs in the small groups database. (Currently, that means that the order of the group is at most 511.) If so, it returns the numbers under which H occurs in the database.

• 

The value returned is a sequence of two numbers such that calling SmallGroup with those two numbers as arguments returns the group H.  The first number is the order of G.

Examples

withGroupTheory:

We identify the three-dimensional projective special linear group over the field of two elements.

IdentifySmallGroupPSL3,2

168,42

(1)

IdentifySmallGroupPSL2,7

168,42

(2)

We see that both groups are isomorphic (because they are both isomorphic to SmallGroup168,42). Now construct a group using the SmallGroup command, then create a Cayley table group that is isomorphic to it, and test that it is still recognized as the same group.

g1SmallGroup96,7

g1 < a permutation group on 96 letters with 6 generators >

(3)

g2CayleyTableGroupg1

g2 < a Cayley table group with 96 elements >

(4)

IdentifySmallGroupg2&comma;assign=iso

96,7

(5)

Domainiso

< a Cayley table group with 96 elements >

(6)

Codomainiso

< a permutation group on 96 letters with 6 generators >

(7)

Using the infolevel facility, we can obtain some information about the progress of the command.

infolevelGroupTheory3

infolevelGroupTheory3

(8)

g3SmallGroup128&comma;1607

g3 < a permutation group on 128 letters with 7 generators >

(9)

IdentifySmallGroupg3

IdentifySmallGroup:   determined group size to be   128
IdentifySmallGroup:   after evaluating hash value, candidates are   {[128, 459, [a1, a2, a3, a4, a5, a6, a7], [[a1, a1, 1/a5], [1/a2, 1/a1, a2, a1, 1/a4], [1/a3, 1/a1, a3, a1], [1/a4, 1/a1, a4, a1], [1/a5, 1/a1, a5, a1], [1/a6, 1/a1, a6, a1], [1/a7, 1/a1, a7, a1], [a2, a2, 1/a6], [1/a3, 1/a2, a3, a2], [1/a4, 1/a2, a4, a2], [1/a5, 1/a2, a5, a2], [1/a6, 1/a2, a6, a2], [1/a7, 1/a2, a7, a2], [a3, a3], [1/a4, 1/a3, a4, a3], [1/a5, 1/a3, a5, a3], [1/a6, 1/a3, a6, a3], [1/a7, 1/a3, a7, a3], [a4, a4], [1/a5, 1/a4, a5, a4], [1/a6, 1/a4, a6, a4], [1/a7, 1/a4, a7, a4], [a5, a5, 1/a7], [1/a6, 1/a5, a6, a5], [1/a7, 1/a5, a7, a5], [a6, a6], [1/a7, 1/a6, a7, a6], [a7, a7]]], [128, 480, [a1, a2, a3, a4, a5, a6, a7], [[a1, a1, 1/a5], [1/a2, 1/a1, a2, a1, 1/a4], [1/a3, 1/a1, a3, a1], [1/a4, 1/a1, a4, a1], [1/a5, 1/a1, a5, a1], [1/a6, 1/a1, a6, a1], [1/a7, 1/a1, a7, a1], [a2, a2, 1/a6], [1/a3, 1/a2, a3, a2], [1/a4, 1/a2, a4, a2], [1/a5, 1/a2, a5, a2], [1/a6, 1/a2, a6, a2], [1/a7, 1/a2, a7, a2], [a3, a3
, 1/a5], [1/a4, 1/a3, a4, a3], [1/a5, 1/a3, a5, a3], [1/a6, 1/a3, a6, a3], [1/a7, 1/a3, a7, a3], [a4, a4], [1/a5, 1/a4, a5, a4], [1/a6, 1/a4, a6, a4], [1/a7, 1/a4, a7, a4], [a5, a5], [1/a6, 1/a5, a6, a5], [1/a7, 1/a5, a7, a5], [a6, a6, 1/a7], [1/a7, 1/a6, a7, a6], [a7, a7]]], [128, 483, [a1, a2, a3, a4, a5, a6, a7], [[a1, a1, 1/a5], [1/a2, 1/a1, a2, a1, 1/a4], [1/a3, 1/a1, a3, a1], [1/a4, 1/a1, a4, a1], [1/a5, 1/a1, a5, a1], [1/a6, 1/a1, a6, a1], [1/a7, 1/a1, a7, a1], [a2, a2, 1/a6], [1/a3, 1/a2, a3, a2], [1/a4, 1/a2, a4, a2], [1/a5, 1/a2, a5, a2], [1/a6, 1/a2, a6, a2], [1/a7, 1/a2, a7, a2], [a3, a3, 1/a5], [1/a4, 1/a3, a4, a3], [1/a5, 1/a3, a5, a3], [1/a6, 1/a3, a6, a3], [1/a7, 1/a3, a7, a3], [a4, a4], [1/a5, 1/a4, a5, a4], [1/a6, 1/a4, a6, a4], [1/a7, 1/a4, a7, a4], [a5, a5, 1/a7], [1/a6, 1/a5, a6, a5], [1/a7, 1/a5, a7, a5], [a6, a6], [1/a7, 1/a6, a7, a6], [a7, a7]]], [128, 1602, [a1, a2, a3, a4, a5, a6, a7], [[a1, a1, 1/a5], [1/a2, 1/a1, a2, a1, 1/a7], [1/a3, 1/a1, a3, a1], [1/a4,
1/a1, a4, a1], [1/a5, 1/a1, a5, a1], [1/a6, 1/a1, a6, a1], [1/a7, 1/a1, a7, a1], [a2, a2, 1/a6], [1/a3, 1/a2, a3, a2], [1/a4, 1/a2, a4, a2], [1/a5, 1/a2, a5, a2], [1/a6, 1/a2, a6, a2], [1/a7, 1/a2, a7, a2], [a3, a3], [1/a4, 1/a3, a4, a3], [1/a5, 1/a3, a5, a3], [1/a6, 1/a3, a6, a3], [1/a7, 1/a3, a7, a3], [a4, a4], [1/a5, 1/a4, a5, a4], [1/a6, 1/a4, a6, a4], [1/a7, 1/a4, a7, a4], [a5, a5, 1/a7], [1/a6, 1/a5, a6, a5], [1/a7, 1/a5, a7, a5], [a6, a6], [1/a7, 1/a6, a7, a6], [a7, a7]]], [128, 1603, [a1, a2, a3, a4, a5, a6, a7], [[a1, a1, 1/a5], [1/a2, 1/a1, a2, a1], [1/a3, 1/a1, a3, a1, 1/a7], [1/a4, 1/a1, a4, a1], [1/a5, 1/a1, a5, a1], [1/a6, 1/a1, a6, a1], [1/a7, 1/a1, a7, a1], [a2, a2, 1/a6], [1/a3, 1/a2, a3, a2], [1/a4, 1/a2, a4, a2], [1/a5, 1/a2, a5, a2], [1/a6, 1/a2, a6, a2], [1/a7, 1/a2, a7, a2], [a3, a3], [1/a4, 1/a3, a4, a3], [1/a5, 1/a3, a5, a3], [1/a6, 1/a3, a6, a3], [1/a7, 1/a3, a7, a3], [a4, a4], [1/a5, 1/a4, a5, a4], [1/a6, 1/a4, a6, a4], [1/a7, 1/a4, a7, a4], [a5, a5, 1/a7],
[1/a6, 1/a5, a6, a5], [1/a7, 1/a5, a7, a5], [a6, a6], [1/a7, 1/a6, a7, a6], [a7, a7]]], [128, 1604, [a1, a2, a3, a4, a5, a6, a7], [[a1, a1, 1/a5], [1/a2, 1/a1, a2, a1], [1/a3, 1/a1, a3, a1], [1/a4, 1/a1, a4, a1], [1/a5, 1/a1, a5, a1], [1/a6, 1/a1, a6, a1], [1/a7, 1/a1, a7, a1], [a2, a2, 1/a6], [1/a3, 1/a2, a3, a2, 1/a7], [1/a4, 1/a2, a4, a2], [1/a5, 1/a2, a5, a2], [1/a6, 1/a2, a6, a2], [1/a7, 1/a2, a7, a2], [a3, a3], [1/a4, 1/a3, a4, a3], [1/a5, 1/a3, a5, a3], [1/a6, 1/a3, a6, a3], [1/a7, 1/a3, a7, a3], [a4, a4], [1/a5, 1/a4, a5, a4], [1/a6, 1/a4, a6, a4], [1/a7, 1/a4, a7, a4], [a5, a5, 1/a7], [1/a6, 1/a5, a6, a5], [1/a7, 1/a5, a7, a5], [a6, a6], [1/a7, 1/a6, a7, a6], [a7, a7]]], [128, 1605, [a1, a2, a3, a4, a5, a6, a7], [[a1, a1, 1/a5], [1/a2, 1/a1, a2, a1], [1/a3, 1/a1, a3, a1], [1/a4, 1/a1, a4, a1, 1/a7], [1/a5, 1/a1, a5, a1], [1/a6, 1/a1, a6, a1], [1/a7, 1/a1, a7, a1], [a2, a2, 1/a6], [1/a3, 1/a2, a3, a2, 1/a7], [1/a4, 1/a2, a4, a2], [1/a5, 1/a2, a5, a2], [1/a6, 1/a2, a6, a2], [1
/a7, 1/a2, a7, a2], [a3, a3], [1/a4, 1/a3, a4, a3], [1/a5, 1/a3, a5, a3], [1/a6, 1/a3, a6, a3], [1/a7, 1/a3, a7, a3], [a4, a4], [1/a5, 1/a4, a5, a4], [1/a6, 1/a4, a6, a4], [1/a7, 1/a4, a7, a4], [a5, a5, 1/a7], [1/a6, 1/a5, a6, a5], [1/a7, 1/a5, a7, a5], [a6, a6], [1/a7, 1/a6, a7, a6], [a7, a7]]], [128, 1606, [a1, a2, a3, a4, a5, a6, a7], [[a1, a1, 1/a5], [1/a2, 1/a1, a2, a1], [1/a3, 1/a1, a3, a1], [1/a4, 1/a1, a4, a1], [1/a5, 1/a1, a5, a1], [1/a6, 1/a1, a6, a1], [1/a7, 1/a1, a7, a1], [a2, a2, 1/a6], [1/a3, 1/a2, a3, a2], [1/a4, 1/a2, a4, a2], [1/a5, 1/a2, a5, a2], [1/a6, 1/a2, a6, a2], [1/a7, 1/a2, a7, a2], [a3, a3], [1/a4, 1/a3, a4, a3, 1/a7], [1/a5, 1/a3, a5, a3], [1/a6, 1/a3, a6, a3], [1/a7, 1/a3, a7, a3], [a4, a4], [1/a5, 1/a4, a5, a4], [1/a6, 1/a4, a6, a4], [1/a7, 1/a4, a7, a4], [a5, a5, 1/a7], [1/a6, 1/a5, a6, a5], [1/a7, 1/a5, a7, a5], [a6, a6], [1/a7, 1/a6, a7, a6], [a7, a7]]], [128, 1607, [a1, a2, a3, a4, a5, a6, a7], [[a1, a1, 1/a5], [1/a2, 1/a1, a2, a1, 1/a7], [1/a3, 1/a1,
a3, a1], [1/a4, 1/a1, a4, a1], [1/a5, 1/a1, a5, a1], [1/a6, 1/a1, a6, a1], [1/a7, 1/a1, a7, a1], [a2, a2, 1/a6], [1/a3, 1/a2, a3, a2], [1/a4, 1/a2, a4, a2], [1/a5, 1/a2, a5, a2], [1/a6, 1/a2, a6, a2], [1/a7, 1/a2, a7, a2], [a3, a3], [1/a4, 1/a3, a4, a3, 1/a7], [1/a5, 1/a3, a5, a3], [1/a6, 1/a3, a6, a3], [1/a7, 1/a3, a7, a3], [a4, a4], [1/a5, 1/a4, a5, a4], [1/a6, 1/a4, a6, a4], [1/a7, 1/a4, a7, a4], [a5, a5, 1/a7], [1/a6, 1/a5, a6, a5], [1/a7, 1/a5, a7, a5], [a6, a6], [1/a7, 1/a6, a7, a6], [a7, a7]]], [128, 1634, [a1, a2, a3, a4, a5, a6, a7], [[a1, a1, 1/a6], [1/a2, 1/a1, a2, a1, 1/a5], [1/a3, 1/a1, a3, a1], [1/a4, 1/a1, a4, a1], [1/a5, 1/a1, a5, a1], [1/a6, 1/a1, a6, a1], [1/a7, 1/a1, a7, a1], [a2, a2, 1/a5], [1/a3, 1/a2, a3, a2], [1/a4, 1/a2, a4, a2], [1/a5, 1/a2, a5, a2], [1/a6, 1/a2, a6, a2], [1/a7, 1/a2, a7, a2], [a3, a3], [1/a4, 1/a3, a4, a3], [1/a5, 1/a3, a5, a3], [1/a6, 1/a3, a6, a3], [1/a7, 1/a3, a7, a3], [a4, a4], [1/a5, 1/a4, a5, a4], [1/a6, 1/a4, a6, a4], [1/a7, 1/a4, a7,
a4], [a5, a5], [1/a6, 1/a5, a6, a5], [1/a7, 1/a5, a7, a5], [a6, a6, 1/a7], [1/a7, 1/a6, a7, a6], [a7, a7]]], [128, 1649, [a1, a2, a3, a4, a5, a6, a7], [[a1, a1, 1/a6], [1/a2, 1/a1, a2, a1, 1/a5], [1/a3, 1/a1, a3, a1], [1/a4, 1/a1, a4, a1], [1/a5, 1/a1, a5, a1], [1/a6, 1/a1, a6, a1], [1/a7, 1/a1, a7, a1], [a2, a2], [1/a3, 1/a2, a3, a2], [1/a4, 1/a2, a4, a2], [1/a5, 1/a2, a5, a2], [1/a6, 1/a2, a6, a2], [1/a7, 1/a2, a7, a2], [a3, a3, 1/a5], [1/a4, 1/a3, a4, a3], [1/a5, 1/a3, a5, a3], [1/a6, 1/a3, a6, a3], [1/a7, 1/a3, a7, a3], [a4, a4], [1/a5, 1/a4, a5, a4], [1/a6, 1/a4, a6, a4], [1/a7, 1/a4, a7, a4], [a5, a5], [1/a6, 1/a5, a6, a5], [1/a7, 1/a5, a7, a5], [a6, a6, 1/a7], [1/a7, 1/a6, a7, a6], [a7, a7]]], [128, 1696, [a1, a2, a3, a4, a5, a6, a7], [[a1, a1, 1/a6], [1/a2, 1/a1, a2, a1, 1/a5], [1/a3, 1/a1, a3, a1], [1/a4, 1/a1, a4, a1], [1/a5, 1/a1, a5, a1], [1/a6, 1/a1, a6, a1], [1/a7, 1/a1, a7, a1], [a2, a2], [1/a3, 1/a2, a3, a2], [1/a4, 1/a2, a4, a2], [1/a5, 1/a2, a5, a2], [1/a6, 1/a2, a6
, a2], [1/a7, 1/a2, a7, a2], [a3, a3, 1/a6], [1/a4, 1/a3, a4, a3], [1/a5, 1/a3, a5, a3], [1/a6, 1/a3, a6, a3], [1/a7, 1/a3, a7, a3], [a4, a4, 1/a5], [1/a5, 1/a4, a5, a4], [1/a6, 1/a4, a6, a4], [1/a7, 1/a4, a7, a4], [a5, a5], [1/a6, 1/a5, a6, a5], [1/a7, 1/a5, a7, a5], [a6, a6, 1/a7], [1/a7, 1/a6, a7, a6], [a7, a7]]], [128, 1720, [a1, a2, a3, a4, a5, a6, a7], [[a1, a1, 1/a6], [1/a2, 1/a1, a2, a1, 1/a5], [1/a3, 1/a1, a3, a1], [1/a4, 1/a1, a4, a1, 1/a5], [1/a5, 1/a1, a5, a1], [1/a6, 1/a1, a6, a1], [1/a7, 1/a1, a7, a1], [a2, a2], [1/a3, 1/a2, a3, a2, 1/a5], [1/a4, 1/a2, a4, a2], [1/a5, 1/a2, a5, a2], [1/a6, 1/a2, a6, a2], [1/a7, 1/a2, a7, a2], [a3, a3], [1/a4, 1/a3, a4, a3], [1/a5, 1/a3, a5, a3], [1/a6, 1/a3, a6, a3], [1/a7, 1/a3, a7, a3], [a4, a4, 1/a5], [1/a5, 1/a4, a5, a4], [1/a6, 1/a4, a6, a4], [1/a7, 1/a4, a7, a4], [a5, a5], [1/a6, 1/a5, a6, a5], [1/a7, 1/a5, a7, a5], [a6, a6, 1/a7], [1/a7, 1/a6, a7, a6], [a7, a7]]]}
IdentifySmallGroup:   testing possible id   459
IdentifySmallGroup:   testing possible id   480
IdentifySmallGroup:   testing possible id   483
IdentifySmallGroup:   testing possible id   1602
IdentifySmallGroup:   testing possible id   1603
IdentifySmallGroup:   testing possible id   1604
IdentifySmallGroup:   testing possible id   1605
IdentifySmallGroup:   testing possible id   1606
IdentifySmallGroup:   testing possible id   1607

128,1607

(10)

Compatibility

• 

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

• 

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

See Also

GroupTheory[AllSmallGroups]

GroupTheory[SmallGroup]