Intersection - Maple Help

GroupTheory

 Intersection
 compute the intersection of two subgroups of a group

 Calling Sequence Intersection( A, B, ... ) A intersect B

Parameters

 A - a permutation group B - a permutation group

Description

 • The Intersection( A, B, ... ) command computes the intersection of one or more permutation groups A, B, ...
 • You can also compute the intersection of two permutation groups A and B by using the intersect operator: A intersect B.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $A≔\mathrm{Group}\left(\mathrm{Perm}\left(\left[\left[1,13,7,24,2,17,4,19\right],\left[3,5,9,23,8,15,10,22\right],\left[6,14,11,21,12,18,16,20\right]\right]\right)\right)$
 ${A}{≔}⟨\left({1}{,}{13}{,}{7}{,}{24}{,}{2}{,}{17}{,}{4}{,}{19}\right)\left({3}{,}{5}{,}{9}{,}{23}{,}{8}{,}{15}{,}{10}{,}{22}\right)\left({6}{,}{14}{,}{11}{,}{21}{,}{12}{,}{18}{,}{16}{,}{20}\right)⟩$ (1)
 > $B≔\mathrm{Group}\left(\mathrm{Perm}\left(\left[\left[1,14,7,21,2,18,4,20\right],\left[3,6,9,11,8,12,10,16\right],\left[5,13,23,24,15,17,22,19\right]\right]\right)\right)$
 ${B}{≔}⟨\left({1}{,}{14}{,}{7}{,}{21}{,}{2}{,}{18}{,}{4}{,}{20}\right)\left({3}{,}{6}{,}{9}{,}{11}{,}{8}{,}{12}{,}{10}{,}{16}\right)\left({5}{,}{13}{,}{23}{,}{24}{,}{15}{,}{17}{,}{22}{,}{19}\right)⟩$ (2)
 > $\mathrm{GroupOrder}\left(A\right),\mathrm{GroupOrder}\left(B\right)$
 ${8}{,}{8}$ (3)
 > $H≔A\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{intersect}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}B$
 ${H}{≔}⟨\left({1}{,}{13}{,}{7}{,}{24}{,}{2}{,}{17}{,}{4}{,}{19}\right)\left({3}{,}{5}{,}{9}{,}{23}{,}{8}{,}{15}{,}{10}{,}{22}\right)\left({6}{,}{14}{,}{11}{,}{21}{,}{12}{,}{18}{,}{16}{,}{20}\right)⟩{\cap }⟨\left({1}{,}{14}{,}{7}{,}{21}{,}{2}{,}{18}{,}{4}{,}{20}\right)\left({3}{,}{6}{,}{9}{,}{11}{,}{8}{,}{12}{,}{10}{,}{16}\right)\left({5}{,}{13}{,}{23}{,}{24}{,}{15}{,}{17}{,}{22}{,}{19}\right)⟩$ (4)
 > $\mathrm{GroupOrder}\left(H\right)$
 ${4}$ (5)
 > $\mathrm{Intersection}\left(H\right)$
 $⟨\left({1}{,}{7}{,}{2}{,}{4}\right)\left({3}{,}{9}{,}{8}{,}{10}\right)\left({5}{,}{23}{,}{15}{,}{22}\right)\left({6}{,}{11}{,}{12}{,}{16}\right)\left({13}{,}{24}{,}{17}{,}{19}\right)\left({14}{,}{21}{,}{18}{,}{20}\right)⟩$ (6)
 > $A≔\mathrm{Group}\left(\mathrm{Perm}\left(\left[\left[2,4\right],\left[3,5\right]\right]\right),\mathrm{Perm}\left(\left[\left[2,4,6\right]\right]\right)\right)$
 ${A}{≔}⟨\left({2}{,}{4}\right)\left({3}{,}{5}\right){,}\left({2}{,}{4}{,}{6}\right)⟩$ (7)
 > $B≔\mathrm{Group}\left(\mathrm{Perm}\left(\left[\left[1,3\right],\left[2,6\right]\right]\right),\mathrm{Perm}\left(\left[\left[2,6,4\right]\right]\right)\right)$
 ${B}{≔}⟨\left({1}{,}{3}\right)\left({2}{,}{6}\right){,}\left({2}{,}{6}{,}{4}\right)⟩$ (8)
 > $C≔\mathrm{Group}\left(\mathrm{Perm}\left(\left[\left[1,3\right],\left[2,4\right]\right]\right),\mathrm{Perm}\left(\left[\left[2,6,4\right]\right]\right)\right)$
 ${C}{≔}⟨\left({1}{,}{3}\right)\left({2}{,}{4}\right){,}\left({2}{,}{6}{,}{4}\right)⟩$ (9)
 > $J≔\mathrm{Intersection}\left(A,B,C\right)$
 ${J}{≔}⟨\left({2}{,}{4}{,}{6}\right)⟩{\cap }⟨\left({1}{,}{3}\right)\left({2}{,}{4}\right){,}\left({2}{,}{6}{,}{4}\right)⟩$ (10)
 > $\mathrm{GroupOrder}\left(J\right)$
 ${3}$ (11)

Compatibility

 • The GroupTheory[Intersection] command was introduced in Maple 17.