IsTransitive - Maple Help
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GroupTheory

 IsTransitive
 determine whether a permutation group is transitive

 Calling Sequence IsTransitive( G, domain )

Parameters

 G - a permutation group domain - (optional) a stable set (or list) of positive integers

Description

 • A permutation group $G$ (acting on the set$\left\{1,2,\dots ,n\right\}$ is transitive if, for any $\mathrm{\alpha }$ and $\mathrm{\beta }$, there is a permutation $g$ in $G$ for which ${\mathrm{\alpha }}^{g}=\mathrm{\beta }$. Alternatively, $G$ is transitive if it has precisely one orbit.
 • The domain argument, which is optional and is, by default, equal to the support of G, specifies a stable set under the action of G on which to test the transitivity of G.
 • The IsTransitive( G ) command returns true if the permutation group G is transitive, and returns false otherwise. The group G must be an instance of a permutation group.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$

The following group is not transitive because it has two orbits.

 > $G≔\mathrm{PermutationGroup}\left(\left\{\mathrm{Perm}\left(\left[\left[1,2\right]\right]\right),\mathrm{Perm}\left(\left[\left[1,2,3\right],\left[4,5\right]\right]\right)\right\}\right)$
 ${G}{≔}⟨\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}\right)⟩$ (1)
 > $\mathrm{IsTransitive}\left(G\right)$
 ${\mathrm{false}}$ (2)
 > $\mathrm{Orbits}\left(G\right)$
 $\left[{{1}}^{⟨\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}\right)⟩}{,}{{4}}^{⟨\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}\right)⟩}\right]$ (3)

However, it is transitive on each of its orbits.

 > $\mathrm{IsTransitive}\left(G,\left\{1,2,3\right\}\right)$
 ${\mathrm{true}}$ (4)
 > $\mathrm{IsTransitive}\left(G,\left[4,5\right]\right)$
 ${\mathrm{true}}$ (5)
 > $G≔\mathrm{PermutationGroup}\left(\left\{\mathrm{Perm}\left(\left[\left[1,3,5\right]\right]\right),\mathrm{Perm}\left(\left[\left[1,5\right]\right]\right)\right\}\right)$
 ${G}{≔}⟨\left({1}{,}{5}\right){,}\left({1}{,}{3}{,}{5}\right)⟩$ (6)
 > $\mathrm{IsTransitive}\left(G\right)$
 ${\mathrm{true}}$ (7)
 > $\mathrm{IsTransitive}\left(G,\left\{1,2,3,4,5\right\}\right)$
 ${\mathrm{false}}$ (8)
 > $\mathrm{IsTransitive}\left(\mathrm{Alt}\left(10\right)\right)$
 ${\mathrm{true}}$ (9)
 > $\mathrm{IsTransitive}\left(\mathrm{FrobeniusGroup}\left(12822,2\right)\right)$
 ${\mathrm{true}}$ (10)

Compatibility

 • The GroupTheory[IsTransitive] command was introduced in Maple 17.
 • For more information on Maple 17 changes, see Updates in Maple 17.