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.

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

$G≔\mathrm{PermutationGroup}\left(\left\{\mathrm{Perm}\left(\left[\left[1\,2\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[1\,2\,3\right]\,\left[4\,5\right]\right]\right)\right\}\right)$

$G≔⟨\left(1\,2\right)\,\left(1\,2\,3\right)\left(4\,5\right)⟩$

$\mathrm{IsTransitive}\left(G\right)$

$\mathrm{false}$

$\mathrm{Orbits}\left(G\right)$

$\left[1^{⟨\left(1\,2\right)\,\left(1\,2\,3\right)\left(4\,5\right)⟩}\,4^{⟨\left(1\,2\right)\,\left(1\,2\,3\right)\left(4\,5\right)⟩}\right]$

$\mathrm{IsTransitive}\left(G\,\left\{1\,2\,3\right\}\right)$

$\mathrm{true}$

$\mathrm{IsTransitive}\left(G\,\left[4\,5\right]\right)$

$\mathrm{true}$

$G≔\mathrm{PermutationGroup}\left(\left\{\mathrm{Perm}\left(\left[\left[1\,5\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[1\,3\,5\right]\right]\right)\right\}\right)$

$G≔⟨\left(1\,3\,5\right)\,\left(1\,5\right)⟩$

$\mathrm{IsTransitive}\left(G\right)$

$\mathrm{true}$

$\mathrm{IsTransitive}\left(G\,\left\{1\,2\,3\,4\,5\right\}\right)$

$\mathrm{false}$

$\mathrm{IsTransitive}\left(\mathrm{Alt}\left(10\right)\right)$

$\mathrm{true}$

$\mathrm{IsTransitive}\left(\mathrm{FrobeniusGroup}\left(12822\,2\right)\right)$

$\mathrm{true}$

The GroupTheory[IsTransitive] command was introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.