group - Maple Help

Overview of the group Package

 Calling Sequence group[command](arguments) command(arguments)

Description

 • Important: The group package has been deprecated. Use the superseding GroupTheory package instead.
 • The group package provides commands for working with finite groups generated by permutations, and for groups defined by finite presentations by generators and defining relators.
 • Permutation groups are constructed by using the permgroup command. Certain commands (e.g., LCS) in the group package apply only to finite permutation groups.
 • Finitely presented groups are created by using the grelgroup command.
 • Each command in the group package can be accessed by using either the long form or the short form of the command name in the command calling sequence.
 • To provide type-checking and conversion routines for the data types used in the group package, the type command accepts disjcyc and the convert command accepts disjcyc and permlist as second arguments. For details, see type/disjcyc and convert/disjcyc.

List of group Package Commands

 • The following is a list of available commands.

 • To display the help page for a particular group command, see Getting Help with a Command in a Package.
 • For help with how to represent groups, subgroups, and group elements for this package, see the information under group[grelgroup], group[permgroup], and group[subgrel].

Examples

Important: The group package has been deprecated. Use the superseding GroupTheory package instead.

An example using the group package to find the order of a permutation group:

 > $\mathrm{with}\left(\mathrm{group}\right):$
 > $\mathrm{grouporder}\left(\mathrm{permgroup}\left(8,\left\{a=\left[\left[1,2\right]\right],b=\left[\left[1,2,3,4,5,6,7,8\right]\right]\right\}\right)\right)$
 ${40320}$ (1)

Find the order, and a Sylow 2-subgroup of one of the Mathieu groups.

 > $a≔\left[\left[1,13\right],\left[2,8\right],\left[3,16\right],\left[4,12\right],\left[6,22\right],\left[7,17\right],\left[9,10\right],\left[11,14\right]\right]:$
 > $b≔\left[\left[1,22,3,21\right],\left[2,18,4,13\right],\left[5,12\right],\left[6,11,7,15\right],\left[8,14,20,10\right],\left[17,19\right]\right]:$
 > $g≔\mathrm{permgroup}\left(23,\left\{a,b\right\}\right):$
 > $n≔\mathrm{grouporder}\left(g\right)$
 ${n}{≔}{443520}$ (2)
 > $\mathrm{ifactors}\left(n\right)$
 $\left[{1}{,}\left[\left[{2}{,}{7}\right]{,}\left[{3}{,}{2}\right]{,}\left[{5}{,}{1}\right]{,}\left[{7}{,}{1}\right]{,}\left[{11}{,}{1}\right]\right]\right]$ (3)

We therefore expect a Sylow 2-subgroup to have order 2^7 = 128.

 > $\mathrm{grouporder}\left(\mathrm{Sylow}\left(g,2\right)\right)$
 ${128}$ (4)