IsDicyclic - Maple Help

GroupTheory

 IsDicyclic
 determine whether a permutation group is a dicyclic group
 IsQuaternion
 determine whether a permutation group is a generalized quaternion group

 Calling Sequence IsDicyclic( G ) IsQuaternionGroup( G )

Parameters

 G - a permutation group

Description

 • The IsDicyclic( G ) command determines whether the finite group G is isomorphic to a dicyclic group of some order, without using an (expensive) isomorphism test. It returns the value true if G is isomorphic to a dicyclic group and returns false otherwise.
 • The IsQuaternionGroup( G ) command returns the value true if the group G is isomorphic to a (generalized) quaternion group, and returns false otherwise.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[2,5,10,16,24,14,27,4\right],\left[7,19,17,26,22,9,15,13\right]\right]\right),\mathrm{Perm}\left(\left[\left[2,13,24,26\right],\left[4,7,16,22\right],\left[5,15,14,17\right],\left[9,27,19,10\right]\right]\right)\right]\right)$
 ${G}{≔}⟨\left({2}{,}{5}{,}{10}{,}{16}{,}{24}{,}{14}{,}{27}{,}{4}\right)\left({7}{,}{19}{,}{17}{,}{26}{,}{22}{,}{9}{,}{15}{,}{13}\right){,}\left({2}{,}{13}{,}{24}{,}{26}\right)\left({4}{,}{7}{,}{16}{,}{22}\right)\left({5}{,}{15}{,}{14}{,}{17}\right)\left({9}{,}{27}{,}{19}{,}{10}\right)⟩$ (1)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${16}$ (2)
 > $\mathrm{IsDicyclic}\left(G\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{IsQuaternionGroup}\left(G\right)$
 ${\mathrm{true}}$ (4)
 > $G≔\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[2,5,10,4\right],\left[7,19,17,15,13\right],\left[14,27,16,24\right]\right]\right),\mathrm{Perm}\left(\left[\left[2,27,10,24\right],\left[4,16,5,14\right],\left[13,19\right],\left[15,17\right]\right]\right)\right]\right)$
 ${G}{≔}⟨\left({2}{,}{5}{,}{10}{,}{4}\right)\left({7}{,}{19}{,}{17}{,}{15}{,}{13}\right)\left({14}{,}{27}{,}{16}{,}{24}\right){,}\left({2}{,}{27}{,}{10}{,}{24}\right)\left({4}{,}{16}{,}{5}{,}{14}\right)\left({13}{,}{19}\right)\left({15}{,}{17}\right)⟩$ (5)
 > $\mathrm{IsDicyclic}\left(G\right)$
 ${\mathrm{true}}$ (6)
 > $\mathrm{IsQuaternionGroup}\left(G\right)$
 ${\mathrm{false}}$ (7)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${40}$ (8)
 > $\mathrm{IsDicyclic}\left(\mathrm{DihedralGroup}\left(18\right)\right)$
 ${\mathrm{false}}$ (9)
 > $\mathrm{map}\left(\mathrm{IsDicyclic},\mathrm{AllSmallGroups}\left(24\right)\right)$
 $\left[{\mathrm{false}}{,}{\mathrm{false}}{,}{\mathrm{false}}{,}{\mathrm{true}}{,}{\mathrm{false}}{,}{\mathrm{false}}{,}{\mathrm{false}}{,}{\mathrm{false}}{,}{\mathrm{false}}{,}{\mathrm{false}}{,}{\mathrm{false}}{,}{\mathrm{false}}{,}{\mathrm{false}}{,}{\mathrm{false}}{,}{\mathrm{false}}\right]$ (10)