Advanced Math - Maple Programming Help

Maple 2017 includes numerous cutting-edge updates in a variety of branches of mathematics.

GroupTheory

The GroupTheory package has been extended and improved in several respects for Maple 2017.  Most notable are an implementation of the Burnside-Dixon-Schneider algorithm to compute the (ordinary) character table of a finite group, and the redesign of the database architecture for small groups that allows for new and more flexible search options for the SearchSmallGroups command.  In addition, the NumGroups command has been improved for this release.  Finally, most objects produced by the GroupTheory package now have custom context-sensitive menus available.

Improvements to the SearchSmallGroups command

You can now search for groups whose specified subgroups and quotients have a designated order, as well as a specific group ID.  Previously, if you wanted to find groups whose center had order equal to 4, you would have to have specified individual searches for each of the (two) groups of order 4 separately and then manually combine the results:

$\mathrm{with}\left(\mathrm{GroupTheory}\right):$$Aâ‰”\mathrm{SearchSmallGroups}\left(\mathrm{order}=1..20,\mathrm{center}=\left[4,1\right]\right)$

 ${A}{≔}\left[{4}{,}{1}\right]{,}\left[{16}{,}{6}\right]{,}\left[{16}{,}{13}\right]$ (1.1.1)

$Bâ‰”\mathrm{SearchSmallGroups}\left(\mathrm{order}=1..20,\mathrm{center}=\left[4,2\right]\right)$

 ${B}{≔}\left[{4}{,}{2}\right]{,}\left[{16}{,}{3}\right]{,}\left[{16}{,}{4}\right]{,}\left[{16}{,}{11}\right]{,}\left[{16}{,}{12}\right]$ (1.1.2)

$A,B$

 $\left[{4}{,}{1}\right]{,}\left[{16}{,}{6}\right]{,}\left[{16}{,}{13}\right]{,}\left[{4}{,}{2}\right]{,}\left[{16}{,}{3}\right]{,}\left[{16}{,}{4}\right]{,}\left[{16}{,}{11}\right]{,}\left[{16}{,}{12}\right]$ (1.1.3)

Now it is possible to simply indicate the order in a single invocation of the command:

$\mathrm{SearchSmallGroups}\left(\mathrm{order}=1..20,\mathrm{center}=4\right)$

 $\left[{4}{,}{1}\right]{,}\left[{4}{,}{2}\right]{,}\left[{16}{,}{3}\right]{,}\left[{16}{,}{4}\right]{,}\left[{16}{,}{6}\right]{,}\left[{16}{,}{11}\right]{,}\left[{16}{,}{12}\right]{,}\left[{16}{,}{13}\right]$ (1.1.4)

This works for all the subgroup and quotient properties that are supported by the SearchSmallGroups command.

In addition, several new properties have been added to the SearchSmallGroups command.  These are: classnumber (the number of conjugacy classes), centralquotient (the order, or group ID of the quotient of the group by its center), fittingquotient (the quotient of the group by its Fitting subgroup) and frattiniquotient (the quotient of the group by its Frattini subgroup).

Computing Character Tables of Finite Groups

Maple 2017 includes a new command, CharacterTable, in the GroupTheory package to compute the ordinary character table of a finite group.

$\mathrm{with}\left(\mathrm{GroupTheory}\right):$$Gâ‰”\mathrm{Alt}\left(4\right)$

 ${G}{≔}{{\mathbf{A}}}_{{4}}$ (1.2.1)

$\mathrm{ct}â‰”\mathrm{CharacterTable}\left(G\right)$

 ${\mathrm{ct}}{≔}\left[\begin{array}{ccccc}{C}& {\mathrm{1a}}& {\mathrm{2a}}& {\mathrm{3a}}& {\mathrm{3b}}\\ {\mathrm{|C|}}& {1}& {3}& {4}& {4}\\ {\mathrm{X1}}& {1}& {1}& {1}& {1}\\ {\mathrm{X2}}& {1}& {1}& {-}\frac{{1}}{{2}}{-}\frac{{I}{}\sqrt{{3}}}{{2}}& {-}\frac{{1}}{{2}}{+}\frac{{I}{}\sqrt{{3}}}{{2}}\\ {\mathrm{X3}}& {1}& {1}& {-}\frac{{1}}{{2}}{+}\frac{{I}{}\sqrt{{3}}}{{2}}& {-}\frac{{1}}{{2}}{-}\frac{{I}{}\sqrt{{3}}}{{2}}\\ {\mathrm{X4}}& {3}& {-1}& {0}& {0}\end{array}\right]$ (1.2.2)

The first line of the character table displays the class labels, while the second row indicates the size of the corresponding conjugacy class.  The characters themselves occupy succeeding rows, with the value of each character on a conjugacy class in the corresponding column of the table.

Character tables are represented as Maple objects with a number of methods.  For example, the CharacterDegrees command returns a list of the degrees of the irreducible characters of the group, along with their multiplicities.$\mathrm{CharacterDegrees}\left(\mathrm{ct}\right)$

 $\left[\left[{1}{,}{3}\right]{,}\left[{3}{,}{1}\right]\right]$ (1.2.3)

The Display method supports the inclusion of additional information about the character table, such as the associated (prime) power maps and the Frobenius-Schur indicator values.

$\mathrm{Display}\left(\mathrm{ct},\mathrm{showpowermaps},\mathrm{showindicator}\right)$

 C 1a 2a 3a 3b |C| 1 3 4 4 ${C}^{2}$ 1a 1a 3b 3a ${C}^{3}$ 1a 2a 1a 1a ${\mathrm{\nu }}_{2}$ 1 $\mathrm{χ__1}$