find where a group is in the small groups database
(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.
We identify the three-dimensional projective special linear group over the field of two elements.
We see that both groups are isomorphic (because they are both isomorphic to SmallGroup⁡168,42). 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 ≔ SmallGroup⁡96,7
g1≔ < a permutation group on 96 letters with 6 generators >
g2 ≔ CayleyTableGroup⁡g1
g2≔ < a Cayley table group with 96 elements >
< a Cayley table group with 96 elements >
< a permutation group on 96 letters with 6 generators >
Using the infolevel facility, we can obtain some information about the progress of the command.
infolevelGroupTheory ≔ 3
g3 ≔ SmallGroup⁡128,1607
g3≔ < a permutation group on 128 letters with 7 generators >
The GroupTheory[IdentifySmallGroup] command was introduced in Maple 17.
For more information on Maple 17 changes, see Updates in Maple 17.
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