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GroupTheory

 AffineSpecialLinearGroup
 construct the affine special linear group as a permutation group

 Calling Sequence AffineSpecialLinearGroup( n, q ) ASL( n, q )

Parameters

 n - a positive integer q - a prime power greater than $1$

Description

 • The affine special linear group $\mathrm{ASL}\left(n,q\right)$ is the semi-direct product of the special linear group $SL\left(n,q\right)$ with the natural module of dimension $n$ over the field with $q$ elements. It is also called the special affine group, and is sometimes denoted by $\mathrm{SA}\left(n,q\right)$.
 • The AffineSpecialLinearGroup command produces a permutation group isomorphic to the group $\mathrm{ASL}\left(n,q\right)$.

Examples

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

Both of the following equivalent commands create a one-dimensional affine special linear group over the field with $2$ elements.

 > $G≔\mathrm{AffineSpecialLinearGroup}\left(1,2\right)$
 ${G}{≔}{\mathbf{ASL}}\left({1}{,}{2}\right)$ (1)
 > $G≔\mathrm{ASL}\left(1,2\right)$
 ${G}{≔}{\mathbf{ASL}}\left({1}{,}{2}\right)$ (2)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${2}$ (3)

It is clearly a cyclic group of order $2$. In fact, the one-dimensional affine special linear groups are all elementary abelian because, the one-dimensional special linear group being trivial, they are isomorphic to the additive groups of their natural modules.

 > $Q≔\mathrm{select}\left(\mathrm{type},\left[\mathrm{seq}\right]\left(2..100\right),'\mathrm{primepower}'\right):$
 > $G≔\mathrm{map2}\left(\mathrm{ASL},1,Q\right):$
 > $\mathrm{map}\left(\mathrm{GroupOrder},G\right)$
 $\left[{2}{,}{3}{,}{4}{,}{5}{,}{7}{,}{8}{,}{9}{,}{11}{,}{13}{,}{16}{,}{17}{,}{19}{,}{23}{,}{25}{,}{27}{,}{29}{,}{31}{,}{32}{,}{37}{,}{41}{,}{43}{,}{47}{,}{49}{,}{53}{,}{59}{,}{61}{,}{64}{,}{67}{,}{71}{,}{73}{,}{79}{,}{81}{,}{83}{,}{89}{,}{97}\right]$ (4)
 > $\mathrm{andmap}\left(\mathrm{IsElementary},G\right)$
 ${\mathrm{true}}$ (5)

The two-dimensional affine special linear group over a field with $2$ elements is isomorphic to another familiar group.

 > $G≔\mathrm{ASL}\left(2,2\right)$
 ${G}{≔}{\mathbf{ASL}}\left({2}{,}{2}\right)$ (6)
 > $\mathrm{AreIsomorphic}\left(G,\mathrm{Symm}\left(4\right)\right)$
 ${\mathrm{true}}$ (7)
 > $G≔\mathrm{ASL}\left(3,3\right)$
 ${G}{≔}{\mathbf{ASL}}\left({3}{,}{3}\right)$ (8)
 > $\mathrm{IsPrimitive}\left(G\right)$
 ${\mathrm{true}}$ (9)
 > $\mathrm{Transitivity}\left(G\right)$
 ${2}$ (10)
 > $S≔\mathrm{Socle}\left(G\right)$
 ${S}{≔}⟨\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}{,}{6}\right)\left({7}{,}{8}{,}{9}\right)\left({10}{,}{11}{,}{12}\right)\left({13}{,}{14}{,}{15}\right)\left({16}{,}{17}{,}{18}\right)\left({19}{,}{20}{,}{21}\right)\left({22}{,}{23}{,}{24}\right)\left({25}{,}{26}{,}{27}\right){,}\left({1}{,}{10}{,}{19}\right)\left({2}{,}{11}{,}{20}\right)\left({3}{,}{12}{,}{21}\right)\left({4}{,}{13}{,}{22}\right)\left({5}{,}{14}{,}{23}\right)\left({6}{,}{15}{,}{24}\right)\left({7}{,}{16}{,}{25}\right)\left({8}{,}{17}{,}{26}\right)\left({9}{,}{18}{,}{27}\right){,}\left({1}{,}{7}{,}{4}\right)\left({2}{,}{8}{,}{5}\right)\left({3}{,}{9}{,}{6}\right)\left({10}{,}{16}{,}{13}\right)\left({11}{,}{17}{,}{14}\right)\left({12}{,}{18}{,}{15}\right)\left({19}{,}{25}{,}{22}\right)\left({20}{,}{26}{,}{23}\right)\left({21}{,}{27}{,}{24}\right)⟩$ (11)
 > $\mathrm{GroupOrder}\left(S\right)$
 ${27}$ (12)
 > $\mathrm{IsElementary}\left(S\right)$
 ${\mathrm{true}}$ (13)
 > $\mathrm{IsRegular}\left(S\right)$
 ${\mathrm{true}}$ (14)

 See Also