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GroupTheory

  

FrobeniusGroup

  

retrieve a group from the database of Frobenius groups

  

NumFrobeniusGroups

  

return the number of Frobenius groups of a given order

  

IdentifyFrobeniusGroup

  

return the database ID of a Frobenius group isomorphic to a given Frobenius group

  

AllFrobeniusGroups

  

retrieve all Frobenius groups of a given order

 

Calling Sequence

Parameters

Options

Description

Examples

Compatibility

Calling Sequence

FrobeniusGroup( n, d )

FrobeniusGroup( [ n, d ] )

NumFrobeniusGroups( n )

IdentifyFrobeniusGroup( G, opts )

AllFrobeniusGroups( n )

Parameters

n

-

a positive integer

d

-

a positive integer

G

-

a Frobenius group isomorphic to one in the database

opts

-

option of the form 'assign' = name

Options

• 

The IdentifyFrobeniusGroup command takes an option of the form 'assign' = iso, where iso is an unassigned name.

Description

• 

The Frobenius groups database contains all the Frobenius groups of order less than or equal to 15000, with certain exceptions. The groups are sorted by their orders and they are listed up to isomorphism; that is, for each of the available orders, a complete and irredundant list of isomorphism type representatives of groups is given. These groups are available as permutation groups, and their Frobenius kernels and Frobenius complements are pre-computed.

• 

The exceptional orders n for which Frobenius groups of order n have not yet been fully computed, and are therefore not complete in the database are: 3072, 3584, 5832, 11264, 12288, 13310, 14580.

• 

For the orders 3072, 11264 and 12288, only those Frobenius groups for which the Frobenius kernel is Abelian are included in the database. (For the other orders listed above, no groups are present in the database.)

• 

The FrobeniusGroup( n, d ) command returns the d-th Frobenius group of order n in the Frobenius groups database.  The value of n must be at most 10000 and not among those exceptions listed above.  The value of d must be less than or equal to the number of Frobenius groups of order n. The syntax FrobeniusGroup( [ n, d ] ) is also accepted.

• 

The NumFrobeniusGroups( n ) command returns the number of Frobenius groups of order n, where n is an integer for which the Frobenius groups of order n are known to Maple. A value of 0 is returned if it is known that there are no Frobenius groups of order n.  If the Frobenius groups of order n are known, then a positive integral value is returned.  Otherwise, an exception is raised.

• 

The IdentifyFrobeniusGroup( G ) command attempts to locate an isomorphic copy of the given Frobenius group G in the database of Frobenius groups. If G is isomorphic to the d-th Frobenius group of order n in the database, then the pair (n, d) is returned. If G is not a Frobenius group, or if the Frobenius groups of the same order as the order of G are not known (in the database), then an exception is raised.

• 

An option of the form 'assign' = iso causes IdentifyFrobeniusGroup to compute an isomorphism from the input Frobenius group to the Frobenius group in the database. If an exception occurs in the presence of this option, then the name iso is not assigned a value.

• 

The algorithm used by IdentifyFrobeniusGroup involves producing a hash value for the group based on evaluating a translate of the order class polynomial of the group at a particular point modulo a suitable prime number. This turns out to be a perfect hash for the Frobenius groups database, so no additional isomorphism tests are required to identify the group. (The specific prime and evaluation point are subject to change, and therefore are not documented.) For this reason, the use of the assign option can add considerably to the cost of identifying groups of larger order, as it requires an additional, explicit isomorphic computation.

• 

The AllFrobeniusGroups( n ) command returns a list of all the Frobenius groups of order n, where n is a positive integer for which the Frobenius groups of order n are known.

Examples

withGroupTheory:

NumFrobeniusGroups72

2

(1)

G1FrobeniusGroup72,1

G12,3,8,9,4,6,7,5,2,8,4,73,9,6,5,2,43,65,97,8,1,2,43,5,76,8,9,1,3,62,5,84,7,9

(2)

IsFrobeniusGroupG1

true

(3)

H1FrobeniusComplementG1

H12,3,8,9,4,6,7,5,2,8,4,73,9,6,5,2,43,65,97,8

(4)

IsMalnormalH1,G1

true

(5)

G2FrobeniusGroup72,2

G22,8,4,73,9,6,5,2,3,4,65,7,9,8,2,43,65,97,8,1,2,43,5,76,8,9,1,3,62,5,84,7,9

(6)

AreIsomorphicG1,G2

false

(7)

IdentifyFrobeniusGroupDihedralGroup21

42,2

(8)

In this case, the given dihedral group is a Frobenius group, but is larger than the groups in the database.

IdentifyFrobeniusGroupDihedralGroup310

Error, (in GroupTheory:-NumFrobeniusGroups) data for Frobenius groups of order 118098 not available

Cyclic groups are not Frobenius groups, so a different exception is raised in this example.

IdentifyFrobeniusGroupCyclicGroup20

Error, (in GroupTheory:-IdentifyFrobeniusGroup) group is not a Frobenius group

IdentifyFrobeniusGroupSuzuki2B22

20,1

(9)

AreIsomorphicDihedralGroup21,FrobeniusGroup42,2

true

(10)

LAllFrobeniusGroups100

L2,8,25,213,6,15,104,20,24,95,18,23,127,19,22,1113,16,17,14,2,253,154,245,236,107,228,219,2011,1912,1813,1714,16,1,2,4,7,11,3,5,8,12,16,6,9,13,17,20,10,14,18,21,23,15,19,22,24,25,1,3,6,10,152,5,9,14,194,8,13,18,227,12,17,21,2411,16,20,23,25,2,4,11,73,6,15,105,13,25,218,20,24,149,22,23,1216,17,19,18,2,113,154,75,256,108,249,2312,2213,2114,2016,1917,18,1,2,4,7,113,5,8,12,166,9,13,17,2010,14,18,21,2315,19,22,24,25,1,3,6,10,152,5,9,14,194,8,13,18,227,12,17,21,2411,16,20,23,25,2,7,11,43,6,15,105,17,25,188,9,24,2312,20,22,1413,19,21,16,2,113,154,75,256,108,249,2312,2213,2114,2016,1917,18,1,2,4,7,113,5,8,12,166,9,13,17,2010,14,18,21,2315,19,22,24,25,1,3,6,10,152,5,9,14,194,8,13,18,227,12,17,21,2411,16,20,23,25

(11)

mapIsFrobeniusGroup,L

true,true,true

(12)

GDihedralGroup333:

Use the assign option to request that an explicit isomorphism be computed.

idIdentifyFrobeniusGroupDihedralGroup333,assign=η

id666,2

(13)

Construct the Frobenius group directly from the database.

F1FrobeniusGroupid:

Construct a group as the image of the computed isomorphism eta.

F2Imageη

F21,165,303,108,254,59,197,11,148,286,129,267,72,218,23,161,307,112,250,93,231,36,182,320,125,271,76,214,57,195,329,146,284,89,235,40,178,21,159,297,110,248,53,199,6,142,318,123,261,74,212,17,163,301,106,282,87,225,38,176,314,127,265,70,246,51,189,330,140,278,91,229,34,210,15,153,299,104,242,55,193,3,174,312,117,263,68,206,19,157,295,138,276,81,227,32,170,316,121,259,102,240,45,191,327,134,280,85,223,66,204,10,155,293,98,244,49,187,30,168,306,119,257,62,208,13,151,326,132,270,83,221,26,172,310,115,291,96,234,47,185,322,136,274,79,255,60,198,12,149,287,100,238,43,219,24,162,308,113,251,64,202,8,183,321,126,272,77,215,28,166,304,147,285,90,236,41,179,324,130,268,111,249,54,200,7,143,289,94,232,75,213,18,164,302,107,253,58,196,39,177,315,128,266,71,217,22,160,331,141,279,92,230,35,181,319,124,300,105,243,56,194,4,145,283,88,264,69,207,20,158,296,109,247,52,228,33,171,317,122,260,73,211,16,192,328,135,281,86,224,37,175,313,156,294,99,245,50,188,5,139,277,120,258,63,209,14,152,298,103,241,84,222,27,173,311,116,262,67,205,48,186,323,137,275,80,226,31,169,333,150,288,101,239,44,190,2,133,309,114,252,65,203,9,154,292,97,273,78,216,29,167,305,118,256,61,237,42,180,325,131,269,82,220,25,201,332,144,290,95,233,46,184,1,3032,2973,2944,2935,2996,2887,2878,2859,28410,28311,29012,28913,27914,27815,27716,27617,27518,27419,28120,28021,30922,27023,26924,26825,26726,26627,26528,27229,27130,30031,26132,26033,25934,25835,25736,25637,26338,26239,29140,25241,25142,25043,24944,24845,24746,25447,25348,28249,24350,24251,24152,24053,23954,23855,24556,24457,27358,23459,23360,23261,23162,23063,22964,23665,23566,26467,22568,22469,22370,22271,22172,22073,22774,22675,25576,21677,21578,21479,21380,21281,21182,21883,21784,24685,20786,20687,20588,20489,20390,20291,20992,20893,23794,19895,19796,19697,19598,19499,193100,200101,199102,228103,189104,188105,187106,186107,185108,184109,191110,190111,219112,180113,179114,178115,177116,176117,175118,182119,181120,210121,171122,170123,169124,168125,167126,166127,173128,172129,201130,162131,161132,160133,159134,158135,157136,164137,163138,192139,153140,152141,151142,150143,149144,148145,155146,154147,183156,174286,332292,329295,328296,327298,330301,323302,322304,321305,320306,319307,325308,324310,315311,314312,313316,317318,333326,331

(14)

Check that F1 and F2 are, in fact, the same group.

IsSubgroupF1,F2andIsSubgroupF2,F1

true

(15)

The smallest insoluble (in fact, perfect) Frobenius group has order 14520.

idSearchFrobeniusGroupssoluble=false

id14520,2

(16)

GFrobeniusGroupid

G2,