ProjectiveGeneralLinearGroup - Maple Help

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GroupTheory

 ProjectiveGeneralLinearGroup
 construct a permutation group isomorphic to a projective general linear group

 Calling Sequence ProjectiveGeneralLinearGroup(n, q) PGL(n, q)

Parameters

 n - a positive integer q - power of a prime number

Description

 • The projective general linear group $PGL\left(n,q\right)$ is the quotient of the general linear group $GL\left(n,q\right)$ by its center.
 • The ProjectiveGeneralLinearGroup( n, q ) command returns a permutation group isomorphic to the projective general linear group $PGL\left(n,q\right)$ . The PGL(n, q) command is provided as an alias.
 • The ranges for n and q are as follows:

 $n=2$ $q\le 100$ $n=3$ $q\le 20$ $n=4$ $q\le 10$ $n=5$ $q\le 5$ $n=6,7,8,9,10$ $q=2$

 • If either of n or q is non-numeric, then a symbolic group is returned.
 • In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{ProjectiveGeneralLinearGroup}\left(2,49\right)$
 ${G}{≔}{\mathbf{PGL}}\left({2}{,}{49}\right)$ (1)
 > $\mathrm{Generators}\left(G\right)$
 $\left[\left({3}{,}{50}{,}{49}{,}{48}{,}{47}{,}{46}{,}{45}{,}{44}{,}{8}{,}{43}{,}{42}{,}{41}{,}{40}{,}{39}{,}{38}{,}{37}{,}{7}{,}{36}{,}{35}{,}{34}{,}{33}{,}{32}{,}{31}{,}{30}{,}{6}{,}{29}{,}{28}{,}{27}{,}{26}{,}{25}{,}{24}{,}{23}{,}{5}{,}{22}{,}{21}{,}{20}{,}{19}{,}{18}{,}{17}{,}{16}{,}{4}{,}{15}{,}{14}{,}{13}{,}{12}{,}{11}{,}{10}{,}{9}\right){,}\left({1}{,}{2}{,}{6}\right)\left({3}{,}{4}{,}{8}\right)\left({9}{,}{25}{,}{33}\right)\left({10}{,}{20}{,}{37}\right)\left({11}{,}{32}{,}{24}\right)\left({12}{,}{15}{,}{41}\right)\left({13}{,}{38}{,}{16}\right)\left({14}{,}{30}{,}{23}\right)\left({17}{,}{31}{,}{19}\right)\left({18}{,}{44}{,}{47}\right)\left({21}{,}{46}{,}{43}\right)\left({22}{,}{39}{,}{49}\right)\left({26}{,}{34}{,}{50}\right)\left({27}{,}{48}{,}{35}\right)\left({28}{,}{42}{,}{40}\right)\left({29}{,}{45}{,}{36}\right)\right]$ (2)
 > $G≔\mathrm{ProjectiveGeneralLinearGroup}\left(5,2\right)$
 ${G}{≔}{\mathbf{PGL}}\left({5}{,}{2}\right)$ (3)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${9999360}$ (4)
 > $\mathrm{GroupOrder}\left(\mathrm{PGL}\left(4,3\right)\right)$
 ${12130560}$ (5)

Several among the small projective general linear groups are isomorphic to some familiar permutation groups.

 > $\mathrm{AreIsomorphic}\left(\mathrm{PGL}\left(2,2\right),\mathrm{Symm}\left(3\right)\right)$
 ${\mathrm{true}}$ (6)
 > $\mathrm{AreIsomorphic}\left(\mathrm{PGL}\left(2,3\right),\mathrm{Symm}\left(4\right)\right)$
 ${\mathrm{true}}$ (7)
 > $\mathrm{AreIsomorphic}\left(\mathrm{PGL}\left(2,4\right),\mathrm{Alt}\left(5\right)\right)$
 ${\mathrm{true}}$ (8)
 > $\mathrm{AreIsomorphic}\left(\mathrm{PGL}\left(2,5\right),\mathrm{Symm}\left(5\right)\right)$
 ${\mathrm{true}}$ (9)
 > $\mathrm{GroupOrder}\left(\mathrm{PGL}\left(3,q\right)\right)$
 $\frac{\left({{q}}^{{3}}{-}{1}\right){}\left({{q}}^{{3}}{-}{q}\right){}\left({{q}}^{{3}}{-}{{q}}^{{2}}\right)}{{q}{-}{1}}$ (10)

Compatibility

 • The GroupTheory[ProjectiveGeneralLinearGroup] command was introduced in Maple 17.