TabulateSimpleGroups - Maple Help

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GroupTheory

 TabulateSimpleGroups
 list finite simple groups

 Calling Sequence TabulateSimpleGroups( r, opts )

Parameters

 r - posint .. posint ; a range of orders to tabulate opts - (optional) equations of the form keyword = value; see Options section

Options

 • cyclic : true or false; (default false)
 • alternating : true or false; (default true)
 • lie : true or false; (default true)
 • minlierank : positive integer; (default 1)
 • psl2 : true or false; (default true)

Description

 • The TabulateSimpleGroups( a .. b ) command lists the finite simple groups whose orders lie in the range a .. b, where a and b are positive integers with a <= b.
 • The output is a sequence of pairs of the form [ n, L ], where n is the order of the group, and L is a list of strings describing simple groups of order n. A simple group might have multiple descriptions, and there may be more than one simple group (up to isomorphism) of a given order. The list L does not attempt to distinguish these cases; all known descriptions of simple groups of order n are listed.
 • You can use the options to selectively include or exclude certain classes of simple groups.
 • The cyclic = true option causes TabulateSimpleGroups to include the groups of prime order in the output. By default, the cyclic option is set to false.
 • The alternating option is true by default, so the alternating groups within the selected range will be included in the output. To exclude alternating groups (other than those that are isomorphic to a group in another class), pass the alternating = false option. You can cause the simple groups of Lie type to be excluded from the output by using the lie = false option; it is true by default.
 • The lie option is true by default, so simple groups of Lie type are included in the output. To exclude the finite simple groups of Lie type, use the lie = false option.
 • The minlierank = r option is used to set the minimum Lie rank of groups included in the output to be r. The default value of minlierank is $1$. This option has no effect if lie = false is set.
 • Since the number of simple groups of the form PSL(2,q) dominates the simple groups occurring within larger ranges, you can use the option psl2 = false to suppress those of this form.
 • Note that excluding a class of groups via options only causes Maple to omit those descriptions for simple groups. If a simple group occurs in another family of simple groups that is not excluded by the option settings passed, then it will be included in the output. For example, the alternating group of degree $8$ can be described equally as the projective special linear group PSL(4,2), and the latter group will be included in the output even though alternating = false was passed. Similarly, the alternating groups of degrees $5$ and $6$ will still appear in the output when the psl2 = false option is passed (unless, of course, alternating = false is also passed).

Examples

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

Note that the simple groups of orders $60$ and $168$ have multiple descriptions, but they are all isomorphic.

 > $\mathrm{TabulateSimpleGroups}\left(1..300\right)$
 $\left[\left[{60}{,}\left[{"Alt\left(5\right)"}{,}{"PSL\left(2,4\right)"}{,}{"PSL\left(2,5\right)"}\right]\right]{,}\left[{168}{,}\left[{"PSL\left(2,7\right)"}{,}{"PSL\left(3,2\right)"}\right]\right]\right]$ (1)

Notice that using the alternating = false option causes "Alt(5)" to not appear in the output, but the group does appear in the alternate guise of "PSL(2,4)" and "PSL(2,5)", all being descriptions of the same group.

 > $\mathrm{TabulateSimpleGroups}\left(1..300,'\mathrm{alternating}'=\mathrm{false}\right)$
 $\left[\left[{60}{,}\left[{"PSL\left(2,4\right)"}{,}{"PSL\left(2,5\right)"}\right]\right]{,}\left[{168}{,}\left[{"PSL\left(2,7\right)"}{,}{"PSL\left(3,2\right)"}\right]\right]\right]$ (2)

Similarly, the two-dimensional projective special linear groups are suppressed by the use of the minlierank = 2 option, but are actually present in other forms as an alternating group, or a linear group of higher degree (and Lie rank).

 > $\mathrm{TabulateSimpleGroups}\left(1..300,'\mathrm{minlierank}'=2\right)$
 $\left[\left[{60}{,}{"Alt\left(5\right)"}\right]{,}\left[{168}{,}{"PSL\left(3,2\right)"}\right]\right]$ (3)

Here, the simple groups of order $20160$ again have multiple descriptions but, in this case, there are in fact two distinct isomorphism classes.

 > $\mathrm{TabulateSimpleGroups}\left(20000..22000\right)$
 $\left[\left[{20160}{,}\left[{"Alt\left(8\right)"}{,}{"PSL\left(3,4\right)"}{,}{"PSL\left(4,2\right)"}\right]\right]\right]$ (4)
 > $\mathrm{TabulateSimpleGroups}\left({10}^{6}..{10}^{7},':-\mathrm{psl2}'=\mathrm{false}\right)$
 $\left[\left[{1451520}{,}\left[{"P\Omega \left(7,2\right)"}{,}{"PSp\left(6,2\right)"}\right]\right]{,}\left[{1814400}{,}{"Alt\left(10\right)"}\right]{,}\left[{1876896}{,}{"PSL\left(3,7\right)"}\right]{,}\left[{3265920}{,}{"PSU\left(4,3\right)"}\right]{,}\left[{4245696}{,}{"G2\left(3\right)"}\right]{,}\left[{4680000}{,}{"PSp\left(4,5\right)"}\right]{,}\left[{5515776}{,}{"PSU\left(3,8\right)"}\right]{,}\left[{5663616}{,}{"PSU\left(3,7\right)"}\right]{,}\left[{6065280}{,}{"PSL\left(4,3\right)"}\right]{,}\left[{9999360}{,}{"PSL\left(5,2\right)"}\right]\right]$ (5)
 > $\mathrm{nops}\left(\mathrm{TabulateSimpleGroups}\left({10}^{6}..{10}^{7}\right)\right)$
 ${41}$ (6)
 > $\mathrm{nops}\left(\mathrm{TabulateSimpleGroups}\left({10}^{4}..{10}^{5},'\mathrm{cyclic}'=\mathrm{true}\right)\right)$
 ${8377}$ (7)

Compatibility

 • The GroupTheory[TabulateSimpleGroups] command was introduced in Maple 2020.