GroupTheory/ChevalleyG2 - Maple Help

GroupTheory

 ChevalleyG2

 Calling Sequence ChevalleyG2( q )

Parameters

 q - : algebraic : an algebraic expression, taken to be a prime power

Description

 • The Chevalley group ${G}_{2}\left(q\right)$ , for a prime power $q$, is a generically simple group of Lie type. The groups ${G}_{2}\left(q\right)$ were studied by Dickson in 1905.
 • The ChevalleyG2( q ) command returns a permutation group isomorphic to the Chevalley group ${G}_{2}\left(q\right)$ , for prime powers $q\le 13$. For non-numeric values of the argument q, or for prime powers $q$ larger than $13$, a symbolic group representing the group ${G}_{2}\left(q\right)$ is returned.
 • Note that the group ${G}_{2}\left(2\right)$ is not simple, but its derived subgroup is simple (isomorphic to the simple unitary group $PSU\left(3,3\right)$ .
 • For values of $q$ for which ${G}_{2}\left(q\right)$ is available as a permutation group, the generating permutations have orders $2$ and $3$ in each case.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{ChevalleyG2}\left(2\right)$
 ${G}{≔}⟨\left({1}{,}{2}\right)\left({3}{,}{5}\right)\left({4}{,}{7}\right)\left({6}{,}{10}\right)\left({8}{,}{12}\right)\left({9}{,}{13}\right)\left({11}{,}{16}\right)\left({14}{,}{20}\right)\left({17}{,}{23}\right)\left({19}{,}{25}\right)\left({21}{,}{28}\right)\left({22}{,}{29}\right)\left({26}{,}{30}\right)\left({27}{,}{31}\right)\left({32}{,}{33}\right)\left({34}{,}{36}\right)\left({35}{,}{37}\right)\left({38}{,}{40}\right)\left({39}{,}{42}\right)\left({41}{,}{45}\right)\left({43}{,}{47}\right)\left({44}{,}{48}\right)\left({46}{,}{50}\right)\left({52}{,}{55}\right)\left({53}{,}{56}\right)\left({57}{,}{59}\right)\left({60}{,}{62}\right)\left({61}{,}{63}\right){,}\left({1}{,}{3}{,}{6}\right)\left({2}{,}{4}{,}{8}\right)\left({5}{,}{9}{,}{14}\right)\left({7}{,}{11}{,}{17}\right)\left({10}{,}{15}{,}{21}\right)\left({12}{,}{18}{,}{24}\right)\left({13}{,}{19}{,}{26}\right)\left({16}{,}{22}{,}{25}\right)\left({20}{,}{27}{,}{32}\right)\left({28}{,}{33}{,}{35}\right)\left({29}{,}{34}{,}{30}\right)\left({36}{,}{38}{,}{41}\right)\left({37}{,}{39}{,}{43}\right)\left({40}{,}{44}{,}{49}\right)\left({42}{,}{46}{,}{51}\right)\left({45}{,}{50}{,}{54}\right)\left({47}{,}{52}{,}{56}\right)\left({48}{,}{53}{,}{57}\right)\left({55}{,}{58}{,}{60}\right)\left({59}{,}{61}{,}{62}\right)⟩$ (1)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${12096}$ (2)
 > $\mathrm{IsSimple}\left(G\right)$
 ${\mathrm{false}}$ (3)
 > $\mathrm{cs}≔\mathrm{CompositionSeries}\left(G\right)$
 ${\mathrm{cs}}{≔}⟨\left({1}{,}{2}\right)\left({3}{,}{5}\right)\left({4}{,}{7}\right)\left({6}{,}{10}\right)\left({8}{,}{12}\right)\left({9}{,}{13}\right)\left({11}{,}{16}\right)\left({14}{,}{20}\right)\left({17}{,}{23}\right)\left({19}{,}{25}\right)\left({21}{,}{28}\right)\left({22}{,}{29}\right)\left({26}{,}{30}\right)\left({27}{,}{31}\right)\left({32}{,}{33}\right)\left({34}{,}{36}\right)\left({35}{,}{37}\right)\left({38}{,}{40}\right)\left({39}{,}{42}\right)\left({41}{,}{45}\right)\left({43}{,}{47}\right)\left({44}{,}{48}\right)\left({46}{,}{50}\right)\left({52}{,}{55}\right)\left({53}{,}{56}\right)\left({57}{,}{59}\right)\left({60}{,}{62}\right)\left({61}{,}{63}\right){,}\left({1}{,}{3}{,}{6}\right)\left({2}{,}{4}{,}{8}\right)\left({5}{,}{9}{,}{14}\right)\left({7}{,}{11}{,}{17}\right)\left({10}{,}{15}{,}{21}\right)\left({12}{,}{18}{,}{24}\right)\left({13}{,}{19}{,}{26}\right)\left({16}{,}{22}{,}{25}\right)\left({20}{,}{27}{,}{32}\right)\left({28}{,}{33}{,}{35}\right)\left({29}{,}{34}{,}{30}\right)\left({36}{,}{38}{,}{41}\right)\left({37}{,}{39}{,}{43}\right)\left({40}{,}{44}{,}{49}\right)\left({42}{,}{46}{,}{51}\right)\left({45}{,}{50}{,}{54}\right)\left({47}{,}{52}{,}{56}\right)\left({48}{,}{53}{,}{57}\right)\left({55}{,}{58}{,}{60}\right)\left({59}{,}{61}{,}{62}\right)⟩{▹}⟨\left({1}{,}{18}{,}{2}{,}{15}\right)\left({3}{,}{27}{,}{32}{,}{10}{,}{9}{,}{29}{,}{17}{,}{7}\right)\left({4}{,}{23}{,}{22}{,}{13}{,}{6}{,}{33}{,}{31}{,}{5}\right)\left({8}{,}{12}{,}{11}{,}{26}{,}{38}{,}{40}{,}{30}{,}{16}\right)\left({14}{,}{35}{,}{52}{,}{59}{,}{47}{,}{46}{,}{36}{,}{25}\right)\left({19}{,}{34}{,}{50}{,}{43}{,}{57}{,}{55}{,}{37}{,}{20}\right)\left({21}{,}{39}{,}{42}{,}{28}\right)\left({41}{,}{45}{,}{44}{,}{61}{,}{62}{,}{60}{,}{63}{,}{48}\right)\left({49}{,}{53}{,}{58}{,}{56}\right)\left({51}{,}{54}\right){,}\left({1}{,}{35}{,}{31}{,}{9}{,}{8}{,}{23}{,}{25}{,}{19}\right)\left({2}{,}{18}{,}{17}{,}{13}{,}{41}{,}{44}{,}{29}{,}{22}\right)\left({3}{,}{24}{,}{4}{,}{21}\right)\left({5}{,}{28}{,}{56}{,}{61}{,}{52}{,}{51}{,}{38}{,}{16}\right)\left({6}{,}{32}{,}{20}{,}{15}{,}{14}{,}{34}{,}{7}{,}{11}\right)\left({10}{,}{43}{,}{46}{,}{33}\right)\left({26}{,}{30}{,}{54}{,}{37}{,}{48}{,}{58}{,}{39}{,}{27}\right)\left({36}{,}{50}{,}{49}{,}{62}{,}{59}{,}{55}{,}{63}{,}{53}\right)\left({40}{,}{57}{,}{60}{,}{47}\right)\left({42}{,}{45}\right)⟩{▹}⟨⟩$ (4)
 > $\mathrm{seq}\left(\mathrm{IsSimple}\left(H\right),H=\mathrm{cs}\right)$
 ${\mathrm{false}}{,}{\mathrm{true}}{,}{\mathrm{false}}$ (5)
 > $\mathrm{ClassifyFiniteSimpleGroup}\left(\mathrm{cs}\left[2\right]\right)$
 $⟨{\mathbf{\text{CFSG:}}}{\text{Steinberg Group}}{}^{{2}}{A}_{{2}}{}\left({3}\right){=}{\mathrm{PSU}}{}\left({3}{,}{3}\right)⟩$ (6)
 > $\mathrm{IsSimple}\left(\mathrm{DerivedSubgroup}\left(G\right)\right)$
 ${\mathrm{true}}$ (7)
 > $G≔\mathrm{ChevalleyG2}\left(7\right):$
 > $\mathrm{GroupOrder}\left(G\right)$
 ${664376138496}$ (8)
 > $\mathrm{IsSimple}\left(G\right)$
 ${\mathrm{true}}$ (9)
 > $\mathrm{ClassNumber}\left(G\right)$
 ${72}$ (10)
 > $G≔\mathrm{ChevalleyG2}\left(13\right):$
 > $\mathrm{GroupOrder}\left(G\right)$
 ${3914077489672896}$ (11)
 > $\mathrm{IsSimple}\left(G\right)$
 ${\mathrm{true}}$ (12)

If the value of the prime power $q$ is too large, or if $q$ is a non-numeric expression, then a symbolic group representing ${G}_{2}\left(q\right)$ is returned.

 > $G≔\mathrm{ChevalleyG2}\left(q\right)$
 ${G}{≔}{{\mathbit{G}}}_{{2}}{}\left({q}\right)$ (13)
 > $\mathrm{Generators}\left(G\right)$
 > $\mathrm{GroupOrder}\left(G\right)$
 ${{q}}^{{6}}{}\left({{q}}^{{6}}{-}{1}\right){}\left({{q}}^{{2}}{-}{1}\right)$ (14)
 > $\mathrm{IsSimple}\left(G\right)$
 $\left\{\begin{array}{cc}{\mathrm{false}}& {q}{=}{2}\\ {\mathrm{true}}& {\mathrm{otherwise}}\end{array}\right\$ (15)
 > $\mathrm{IsSoluble}\left(G\right)$
 ${\mathrm{false}}$ (16)
 Compatibility