 GroupTheory/Ree2G2 - Maple Help

GroupTheory

 Ree2G2 Calling Sequence Ree2G2( q ) Parameters

 q - : {posint,algebraic} : an odd power of $2$, or an expression Description

 • The Ree groups ${}^{2}G_{2}\left(q\right)$ , for an odd power $q$ of $3$, are a series of (typically) simple groups of Lie type, first constructed by R. Ree. They are defined only for $q={3}^{2e+1}$ an odd power of $3$ (where, here, $0\le e$).
 • The Ree2G2( q ) command constructs a permutation group isomorphic to ${}^{2}G_{2}\left(q\right)$ , for q equal to either $3$ or $27$.
 • If the argument q is not numeric, or if it is an odd power of $3$ greater than $27$, then a symbolic group representing ${}^{2}G_{2}\left(q\right)$ is returned.
 • The Ree group ${}^{2}G_{2}\left(3\right)$ is not simple, but mRee( q ) is simple for admissible values of $3. The derived subgroup of ${}^{2}G_{2}\left(3\right)$ is simple, isomorphic to the group $PSL\left(2,8\right)$ . Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Ree2G2}\left(3\right)$
 ${G}{≔}{}^{{2}}{\mathbit{G}}_{{2}}{}\left({3}\right)$ (1)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${1512}$ (2)
 > $\mathrm{IsSimple}\left(G\right)$
 ${\mathrm{false}}$ (3)
 > $L≔\mathrm{DerivedSubgroup}\left(G\right)$
 ${L}{≔}\left[{}^{{2}}{\mathbit{G}}_{{2}}{}\left({3}\right){,}{}^{{2}}{\mathbit{G}}_{{2}}{}\left({3}\right)\right]$ (4)
 > $\mathrm{IsSimple}\left(L\right)$
 ${\mathrm{true}}$ (5)
 > $\mathrm{ClassifyFiniteSimpleGroup}\left(L\right)$
 $⟨{\mathbf{\text{CFSG:}}}{\text{Chevalley Group}}{{A}}_{{1}}{}\left({8}\right){=}{\mathrm{PSL}}{}\left({2}{,}{8}\right)⟩$ (6)
 > $G≔\mathrm{Ree2G2}\left(27\right)$
 ${G}{≔}{}^{{2}}{\mathbit{G}}_{{2}}{}\left({27}\right)$ (7)
 > $\mathrm{IsSimple}\left(G\right)$
 ${\mathrm{true}}$ (8)
 > $\mathbf{use}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{GraphTheory}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{DrawGraph}\left(\mathrm{GruenbergKegelGraph}\left(G\right)\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end use}$ Currently, the group ${}^{2}G_{2}\left(\mathrm{243}\right)$ (and those for larger odd powers of $3$) are available only as symbolic groups.

 > $G≔\mathrm{Ree2G2}\left(243\right)$
 ${G}{≔}{}^{{2}}{\mathbit{G}}_{{2}}{}\left({243}\right)$ (9)
 > $\mathrm{Generators}\left(G\right)$

Nevertheless, Maple has some knowledge of this group.

 > $\mathrm{GroupOrder}\left(G\right)$
 ${49825657439340552}$ (10)
 > $\mathrm{IsSimple}\left(G\right)$
 ${\mathrm{true}}$ (11)
 > $\mathrm{MinPermRepDegree}\left(G\right)$
 ${14348908}$ (12)

Likewise, for non-numeric values of the argument q, a symbolic group is returned.

 > $\mathrm{IsSimple}\left(\mathrm{Ree2G2}\left(q\right)\right)$
 $\left\{\begin{array}{cc}{\mathrm{false}}& {q}{=}{3}\\ {\mathrm{true}}& {\mathrm{otherwise}}\end{array}\right\$ (13)
 > $\mathrm{ClassNumber}\left(\mathrm{Ree2G2}\left(q\right)\right)$
 ${q}{+}{8}$ (14) Compatibility