IdentifySmallGroup - Maple Help

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GroupTheory

 IdentifySmallGroup
 find where a group is in the small groups database

 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

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$

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

 > $\mathrm{IdentifySmallGroup}\left(\mathrm{PSL}\left(3,2\right)\right)$
 ${168}{,}{42}$ (1)
 > $\mathrm{IdentifySmallGroup}\left(\mathrm{PSL}\left(2,7\right)\right)$
 ${168}{,}{42}$ (2)

We see that both groups are isomorphic (because they are both isomorphic to $\mathrm{SmallGroup}\left(168,42\right)$). Now construct a group using the $\mathrm{SmallGroup}$ command, then create a Cayley table group that is isomorphic to it, and test that it is still recognized as the same group.

 > $\mathrm{g1}≔\mathrm{SmallGroup}\left(96,7\right)$
 ${\mathrm{g1}}{≔}{\mathrm{< a permutation group on 96 letters with 6 generators >}}$ (3)
 > $\mathrm{g2}≔\mathrm{CayleyTableGroup}\left(\mathrm{g1}\right)$
 ${\mathrm{g2}}{≔}{\mathrm{< a Cayley table group with 96 elements >}}$ (4)
 > $\mathrm{IdentifySmallGroup}\left(\mathrm{g2},'\mathrm{assign}=\mathrm{iso}'\right)$
 ${96}{,}{7}$ (5)
 > $\mathrm{Domain}\left(\mathrm{iso}\right)$
 ${\mathrm{< a Cayley table group with 96 elements >}}$ (6)
 > $\mathrm{Codomain}\left(\mathrm{iso}\right)$
 ${\mathrm{< a permutation group on 96 letters with 6 generators >}}$ (7)

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

 > $\mathrm{infolevel}\left[\mathrm{GroupTheory}\right]≔3$
 ${{\mathrm{infolevel}}}_{{\mathrm{GroupTheory}}}{≔}{3}$ (8)
 > $\mathrm{g3}≔\mathrm{SmallGroup}\left(128,1607\right)$
 ${\mathrm{g3}}{≔}{\mathrm{< a permutation group on 128 letters with 7 generators >}}$ (9)
 > $\mathrm{IdentifySmallGroup}\left(\mathrm{g3}\right)$
 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