GroupTheory
IdentifySmallGroup
find where a group is in the small groups database
Calling Sequence
Parameters
Options
Description
Examples
Compatibility
IdentifySmallGroup(G, opts)
G
-
a group
opts
(optional) equations of the form keyword = value, listed below
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.
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.
withGroupTheory:
We identify the three-dimensional projective special linear group over the field of two elements.
IdentifySmallGroupPSL3,2
168,42
IdentifySmallGroupPSL2,7
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.
g1≔SmallGroup96,7
g1≔1,23,214,305,156,147,178,169,1810,2011,1912,2313,2722,5124,4725,4626,5228,6629,6531,4532,4833,8034,7935,5836,5737,5638,5539,6240,6141,6042,5943,6444,6349,7050,6953,7854,7767,9468,9371,9072,8973,8874,8775,9676,9581,8682,8583,9284,91,1,3,22,32,8,26,5,23,9,27,6,24,7,25,31,42,12,45,52,17,48,14,21,18,30,15,46,16,47,51,1310,28,67,83,41,75,35,69,43,77,37,71,39,73,81,3311,29,68,84,42,76,36,70,44,78,38,72,40,74,82,3419,49,85,95,61,91,55,65,63,79,57,87,59,89,93,5320,50,86,96,62,92,56,66,64,80,58,88,60,90,94,54,1,5,7,22,9,31,8,62,14,16,45,18,51,17,153,23,25,32,27,4,26,2410,35,39,67,43,81,41,3711,36,40,68,44,82,42,3812,21,47,52,30,13,48,4619,55,59,85,63,93,61,5720,56,60,86,64,94,62,5828,69,73,83,77,33,75,7129,70,74,84,78,34,76,7249,65,89,95,79,53,91,8750,66,90,96,80,54,92,88,1,7,9,82,16,18,173,25,27,264,24,23,325,22,31,610,39,43,4111,40,44,4212,47,30,4813,46,21,5214,45,51,1519,59,63,6120,60,64,6228,73,77,7529,74,78,7633,71,69,8334,72,70,8435,67,81,3736,68,82,3849,89,79,9150,90,80,9253,87,65,9554,88,66,9655,85,93,5756,86,94,58,1,92,183,274,235,316,227,810,4311,4412,3013,2114,5115,4516,1719,6320,6424,3225,2628,7729,7833,6934,7035,8136,8237,6738,6839,4140,4246,5247,4849,7950,8053,6554,6655,9356,9457,8558,8659,6160,6271,8372,8473,7574,7687,9588,9689,9190,92,1,10,112,19,203,28,294,33,345,35,366,37,387,39,408,41,429,43,4412,49,5013,53,5414,55,5615,57,5816,59,6017,61,6218,63,6421,65,6622,67,6823,69,7024,71,7225,73,7426,75,7627,77,7830,79,8031,81,8232,83,8445,85,8646,87,8847,89,9048,91,9251,93,9452,95,96
g2≔CayleyTableGroupg1
g2≔ < a Cayley table group with 96 elements >
IdentifySmallGroupg2,assign=iso
96,7
Domainiso
< a Cayley table group with 96 elements >
Codomainiso
1,23,214,305,156,147,178,169,1810,2011,1912,2313,2722,5124,4725,4626,5228,6629,6531,4532,4833,8034,7935,5836,5737,5638,5539,6240,6141,6042,5943,6444,6349,7050,6953,7854,7767,9468,9371,9072,8973,8874,8775,9676,9581,8682,8583,9284,91,1,3,22,32,8,26,5,23,9,27,6,24,7,25,31,42,12,45,52,17,48,14,21,18,30,15,46,16,47,51,1310,28,67,83,41,75,35,69,43,77,37,71,39,73,81,3311,29,68,84,42,76,36,70,44,78,38,72,40,74,82,3419,49,85,95,61,91,55,65,63,79,57,87,59,89,93,5320,50,86,96,62,92,56,66,64,80,58,88,60,90,94,54,1,5,7,22,9,31,8,62,14,16,45,18,51,17,153,23,25,32,27,4,26,2410,35,39,67,43,81,41,3711,36,40,68,44,82,42,3812,21,47,52,30,13,48,4619,55,59,85,63,93,61,5720,56,60,86,64,94,62,5828,69,73,83,77,33,75,7129,70,74,84,78,34,76,7249,65,89,95,79,53,91,8750,66,90,96,80,54,92,88,1,7,9,82,16,18,173,25,27,264,24,23,325,22,31,610,39,43,4111,40,44,4212,47,30,4813,46,21,5214,45,51,1519,59,63,6120,60,64,6228,73,77,7529,74,78,7633,71,69,8334,72,70,8435,67,81,3736,68,82,3849,89,79,9150,90,80,9253,87,65,9554,88,66,9655,85,93,5756,86,94,58,1,92,183,274,235,316,227,810,4311,4412,3013,2114,5115,4516,1719,6320,6424,3225,2628,7729,7833,6934,7035,8136,8237,6738,6839,4140,4246,5247,4849,7950,8053,6554,6655,9356,9457,8558,8659,6160,6271,8372,8473,7574,7687,9588,9689,9190,92,1,10,112,19,203,28,294,33,345,35,366,37,387,39,408,41,429,43,4412,49,5013,53,5414,55,5615,57,5816,59,6017,61,6218,63,6421,65,6622,67,6823,69,7024,71,7225,73,7426,75,7627,77,7830,79,8031,81,8232,83,8445,85,8646,87,8847,89,9048,91,9251,93,9452,95,96
Using the infolevel facility, we can obtain some information about the progress of the command.
infolevelGroupTheory≔3
g3≔SmallGroup128,1607
g3≔1,2,8,16,11,18,9,34,24,28,19,30,12,29,255,31,35,20,37,13,36,326,14,39,56,42,58,40,217,15,44,60,47,62,45,2210,17,48,63,50,64,49,2326,73,78,65,80,51,79,7527,74,82,66,84,52,83,7633,85,90,67,92,53,91,8734,86,94,68,96,54,95,8838,55,97,71,43,59,98,6941,57,100,112,102,113,101,7046,61,104,115,106,116,105,7277,108,123,117,81,107,124,11889,110,125,119,93,109,126,12099,111,127,122,103,114,128,121,1,4,10,52,12,17,133,19,23,206,26,41,337,27,46,348,28,48,359,29,49,3611,30,50,3714,51,57,5315,52,61,5416,25,63,3218,24,64,3121,65,70,6722,66,72,6838,77,99,8939,78,100,9040,79,101,9142,80,102,9243,81,103,9344,82,104,9445,83,105,9547,84,106,9655,107,111,10956,75,112,8758,73,113,8559,108,114,11060,76,115,8862,74,116,8669,117,121,11971,118,122,12097,123,127,12598,124,128,126,1,62,143,214,265,337,438,399,4010,4111,4212,5113,5315,5916,5617,5718,5819,6520,6722,7123,7024,7325,7527,8128,7829,7930,8031,8532,8734,9335,9036,9137,9238,4744,9845,9746,10348,10049,10150,10252,10854,11055,6260,6961,11463,11264,11366,11868,12072,12274,10776,11777,8482,12483,12386,10988,11989,9694,12695,12599,106104,128105,127111,116115,121,1,72,153,224,275,346,388,449,4510,4611,4712,5213,5414,5516,6017,6118,6219,6620,6821,6923,7224,7425,7626,7728,8229,8330,8431,8632,8833,8935,9436,9537,9639,9740,9841,9942,4348,10449,10550,10651,10753,10956,7157,11158,5963,11564,11665,11767,11970,12173,10875,11878,12379,12480,8185,11087,12090,12591,12692,93100,127101,128102,103112,122113,114,1,8,11,92,16,18,34,28,30,295,35,37,366,39,42,407,44,47,4510,48,50,4912,25,24,1913,32,31,2014,56,58,2115,60,62,2217,63,64,2326,78,80,7927,82,84,8333,90,92,9134,94,96,9538,97,43,9841,100,102,10146,104,106,10551,75,73,6552,76,74,6653,87,85,6754,88,86,6855,71,59,6957,112,113,7061,115,116,7277,123,81,12489,125,93,12699,127,103,128107,118,108,117109,120,110,119111,122,114,121,1,102,173,234,56,417,468,489,4911,5012,1314,5715,6116,6318,6419,2021,7022,7224,3125,3226,3327,3428,3529,3630,3738,9939,10040,10142,10243,10344,10445,10547,10651,5352,5455,11156,11258,11359,11460,11562,11665,6766,6869,12171,12273,8574,8675,8776,8877,8978,9079,9180,9281,9382,9483,9584,9697,12798,128107,109108,110117,119118,120123,125124,126,1,112,183,164,305,376,427,478,910,5012,2413,3114,5815,6217,6419,2520,3221,5622,6023,6326,8027,8428,2933,9234,9635,3638,4339,4041,10244,4546,10648,4951,7352,7453,8554,8655,5957,11361,11665,7566,7667,8768,8869,7170,11272,11577,8178,7982,8389,9390,9194,9597,9899,103100,101104,105107,108109,110111,114117,118119,120121,122123,124125,126127,128
IdentifySmallGroupg3
128,1607
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]
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