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GroupTheory

 AffineGeneralLinearGroup
 construct the affine general linear group as a permutation group

 Calling Sequence AffineGeneralLinearGroup( n, q ) AGL( n, q )

Parameters

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

Description

 • The affine general linear group $\mathrm{AGL}\left(n,q\right)$ is the semi-direct product of the general linear group $GL\left(n,q\right)$ with the natural module of dimension $n$ over the field with $q$ elements.
 • The AffineGeneralLinearGroup command produces a permutation group isomorphic to the group $\mathrm{AGL}\left(n,q\right)$.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{AffineGeneralLinearGroup}\left(1,2\right)$
 ${G}{≔}{\mathbf{AGL}}\left({1}{,}{2}\right)$ (1)
 > $G≔\mathrm{AGL}\left(1,2\right)$
 ${G}{≔}{\mathbf{AGL}}\left({1}{,}{2}\right)$ (2)
 > $G≔\mathrm{AGL}\left(1,3\right)$
 ${G}{≔}{\mathbf{AGL}}\left({1}{,}{3}\right)$ (3)
 > $\mathrm{AreIsomorphic}\left(G,\mathrm{Symm}\left(3\right)\right)$
 ${\mathrm{true}}$ (4)
 > $G≔\mathrm{AGL}\left(1,4\right)$
 ${G}{≔}{\mathbf{AGL}}\left({1}{,}{4}\right)$ (5)
 > $\mathrm{AreIsomorphic}\left(G,\mathrm{Alt}\left(4\right)\right)$
 ${\mathrm{true}}$ (6)
 > $G≔\mathrm{AGL}\left(1,5\right)$
 ${G}{≔}{\mathbf{AGL}}\left({1}{,}{5}\right)$ (7)
 > $\mathrm{IsFrobeniusGroup}\left(G\right)$
 ${\mathrm{true}}$ (8)
 > $\mathrm{PermGroupRank}\left(G\right)$
 ${2}$ (9)
 > $G≔\mathrm{AGL}\left(2,2\right)$
 ${G}{≔}{\mathbf{AGL}}\left({2}{,}{2}\right)$ (10)
 > $\mathrm{AreIsomorphic}\left(G,\mathrm{Symm}\left(4\right)\right)$
 ${\mathrm{true}}$ (11)
 > $G≔\mathrm{AGL}\left(2,3\right)$
 ${G}{≔}{\mathbf{AGL}}\left({2}{,}{3}\right)$ (12)
 > $\mathrm{Transitivity}\left(G\right)$
 ${2}$ (13)
 > $\mathrm{PermGroupRank}\left(G\right)$
 ${2}$ (14)
 > $S≔\mathrm{Stabilizer}\left(1,G\right)$
 ${S}{≔}⟨\left({2}{,}{6}\right)\left({3}{,}{8}\right)\left({5}{,}{9}\right){,}\left({2}{,}{9}\right)\left({3}{,}{5}\right)\left({6}{,}{8}\right){,}\left({2}{,}{5}{,}{6}{,}{7}{,}{3}{,}{9}{,}{8}{,}{4}\right)⟩$ (15)
 > $\mathrm{AreIsomorphic}\left(S,\mathrm{GL}\left(2,3\right)\right)$
 ${\mathrm{true}}$ (16)
 > $G≔\mathrm{AGL}\left(3,2\right)$
 ${G}{≔}{\mathbf{AGL}}\left({3}{,}{2}\right)$ (17)
 > $\mathrm{IsPrimitive}\left(G\right)$
 ${\mathrm{true}}$ (18)
 > $\mathrm{EARNS}\left(G\right)$
 $⟨\left({1}{,}{5}\right)\left({2}{,}{6}\right)\left({3}{,}{7}\right)\left({4}{,}{8}\right){,}\left({1}{,}{3}\right)\left({2}{,}{4}\right)\left({5}{,}{7}\right)\left({6}{,}{8}\right){,}\left({1}{,}{2}\right)\left({3}{,}{4}\right)\left({5}{,}{6}\right)\left({7}{,}{8}\right)⟩$ (19)
 > $\mathrm{Transitivity}\left(G\right)$
 ${3}$ (20)
 > $\mathrm{AreIsomorphic}\left(\mathrm{Stabilizer}\left(1,G\right),\mathrm{GL}\left(3,2\right)\right)$
 ${\mathrm{true}}$ (21)
 > $G≔\mathrm{AGL}\left(3,3\right)$
 ${G}{≔}{\mathbf{AGL}}\left({3}{,}{3}\right)$ (22)
 > $\mathrm{Transitivity}\left(G\right)$
 ${2}$ (23)
 > $\mathrm{PermGroupRank}\left(G\right)$
 ${2}$ (24)
 > $\mathrm{GroupOrder}\left(\mathrm{Stabilizer}\left(1,G\right)\right)=\mathrm{GroupOrder}\left(\mathrm{GL}\left(3,3\right)\right)$
 ${11232}{=}{11232}$ (25)
 > $\mathrm{EARNS}\left(G\right)$
 $⟨\left({1}{,}{19}{,}{10}\right)\left({2}{,}{20}{,}{11}\right)\left({3}{,}{21}{,}{12}\right)\left({4}{,}{22}{,}{13}\right)\left({5}{,}{23}{,}{14}\right)\left({6}{,}{24}{,}{15}\right)\left({7}{,}{25}{,}{16}\right)\left({8}{,}{26}{,}{17}\right)\left({9}{,}{27}{,}{18}\right){,}\left({1}{,}{3}{,}{2}\right)\left({4}{,}{6}{,}{5}\right)\left({7}{,}{9}{,}{8}\right)\left({10}{,}{12}{,}{11}\right)\left({13}{,}{15}{,}{14}\right)\left({16}{,}{18}{,}{17}\right)\left({19}{,}{21}{,}{20}\right)\left({22}{,}{24}{,}{23}\right)\left({25}{,}{27}{,}{26}\right){,}\left({1}{,}{4}{,}{7}\right)\left({2}{,}{5}{,}{8}\right)\left({3}{,}{6}{,}{9}\right)\left({10}{,}{13}{,}{16}\right)\left({11}{,}{14}{,}{17}\right)\left({12}{,}{15}{,}{18}\right)\left({19}{,}{22}{,}{25}\right)\left({20}{,}{23}{,}{26}\right)\left({21}{,}{24}{,}{27}\right)⟩$ (26)