SpecialLinearGroup - Maple Help

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GroupTheory

 SpecialLinearGroup
 construct a permutation group isomorphic to a special linear group

 Calling Sequence SpecialLinearGroup(n, q) SL(n, q)

Parameters

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

Description

 • The special linear group $SL\left(n,q\right)$ is the set of all $n×n$ nonsingular matrices over a finite field of size $q$ whose determinant is $1$.
 • The SpecialLinearGroup( n, q ) command returns a permutation group isomorphic to the special linear group $SL\left(n,q\right)$ for values of n and q in the implemented ranges.
 • The implemented ranges for the parameters 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 or both of the parameters n and q is non-numeric, then a symbolic group representing the indicated special linear 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

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{SpecialLinearGroup}\left(3,2\right)$
 ${\mathbf{SL}}\left({3}{,}{2}\right)$ (1)
 > $G≔\mathrm{SL}\left(2,3\right)$
 ${G}{≔}{\mathbf{SL}}\left({2}{,}{3}\right)$ (2)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${24}$ (3)
 > $\mathrm{GroupOrder}\left(\mathrm{SL}\left(4,q\right)\right)$
 ${{q}}^{{6}}{}\left({{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{3}}{-}{1}\right){}\left({{q}}^{{4}}{-}{1}\right)$ (4)
 > $G≔\mathrm{SL}\left(2,5\right)$
 ${G}{≔}{\mathbf{SL}}\left({2}{,}{5}\right)$ (5)
 > $\mathrm{IsPerfect}\left(G\right)$
 ${\mathrm{true}}$ (6)

Compatibility

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