A permutation group $G$ (acting on the set$\left\{1\,2\,\dots \,n\right\}$ is semi-regular if the stabilizer of any point is trivial. If, in addition, $G$ acts transitively, then it is said to be regular. This means that the action of $G$ is permutation isomorphic  to the action of $G$ on itself by (right) translation.

Every Abelian transitive permutation group is regular.

The IsRegular( G ) command returns true if the permutation group G is regular, and returns false otherwise. The IsSemiRegular( G ) command returns true if the permutation group G is semi-regular, and returns false otherwise. The group G must be an instance of a permutation group.

The dom option can be used to specify the domain on which G acts.

For regularity in the sense of P. Hall, for groups of prime power order, see GroupTheory[IsRegularPGroup].

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

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

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

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

$\mathrm{false}$

$\mathrm{IsRegular}\left(\mathrm{CyclicGroup}\left(6\right)\right)$

$\mathrm{true}$

$\mathrm{IsRegular}\left(\mathrm{CyclicGroup}\left(6\,':-\mathrm{mindegree}'\right)\right)$

$\mathrm{false}$

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

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

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

$\mathrm{true}$

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

$\mathrm{false}$

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

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

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

$\mathrm{false}$

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

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

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

$\mathrm{false}$

$\mathrm{IsSemiRegular}\left(\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[2\,3\,4\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[5\,6\,7\right]\right]\right)\right]\right)\right)$

$\mathrm{false}$

$G≔\mathrm{Group}\left(\left\{\mathrm{Perm}\left(\left[\left[1\,2\,3\right]\,\left[4\,5\,6\right]\right]\right)\right\}\right)$

$G≔⟨\left(1\,2\,3\right)\left(4\,5\,6\right)⟩$

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

$\mathrm{true}$

$G≔\mathrm{Group}\left(\left\{\mathrm{Perm}\left(\left[\left[1\,2\,3\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[4\,5\,6\right]\right]\right)\right\}\right)$

$G≔⟨\left(1\,2\,3\right)\,\left(4\,5\,6\right)⟩$

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

$\mathrm{false}$

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

$\mathrm{true}$

$\mathrm{IsRegular}\left(G\,\left\{4\,5\,6\right\}\right)$

$\mathrm{true}$

$\mathrm{IsRegular}\left(\mathrm{Symm}\left(3\right)\right)$

$\mathrm{false}$

$\mathrm{IsRegular}\left(\mathrm{TransitiveGroup}\left(6\,2\right)\right)$

$\mathrm{true}$

$\mathrm{AreIsomorphic}\left(\mathrm{Symm}\left(3\right)\,\mathrm{TransitiveGroup}\left(6\,2\right)\right)$

$\mathrm{true}$

$\mathrm{AreIsomorphic}\left(\mathrm{TransitiveGroup}\left(8\,5\right)\,\mathrm{QuaternionGroup}\left(\right)\right)$

$\mathrm{true}$

$\mathrm{IsRegular}\left(\mathrm{TransitiveGroup}\left(8\,5\right)\right)$

$\mathrm{true}$

$G≔\mathrm{DirectProduct}\left(\mathrm{TransitiveGroup}\left(8\,5\right)\,\mathrm{TransitiveGroup}\left(8\,5\right)\right)$

$G≔⟨\left(1\,2\,3\,8\right)\left(4\,5\,6\,7\right)\,\left(1\,7\,3\,5\right)\left(2\,6\,8\,4\right)\,\left(9\,10\,11\,16\right)\left(12\,13\,14\,15\right)\,\left(9\,15\,11\,13\right)\left(10\,14\,16\,12\right)⟩$

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

$\mathrm{false}$

$\mathrm{gens}≔\mathrm{Generators}\left(G\right)\:$

$H≔\mathrm{Subgroup}\left(\left\{\mathrm{gens}\left[1\right]·\mathrm{gens}\left[3\right]\,\mathrm{gens}\left[2\right]·\mathrm{gens}\left[4\right]\right\}\,G\right)$

$H≔⟨\left(1\,2\,3\,8\right)\left(4\,5\,6\,7\right)\left(9\,10\,11\,16\right)\left(12\,13\,14\,15\right)\,\left(1\,7\,3\,5\right)\left(2\,6\,8\,4\right)\left(9\,15\,11\,13\right)\left(10\,14\,16\,12\right)⟩$

$\mathrm{AreIsomorphic}\left(H\,\mathrm{QuaternionGroup}\left(\right)\right)$

$\mathrm{true}$

$\mathrm{IsSemiRegular}\left(H\right)$

$\mathrm{true}$

$\mathrm{IsRegular}\left(\mathrm{FrobeniusGroup}\left(15000\,3\right)\right)$

$\mathrm{false}$

$\mathrm{it}≔\mathrm{AllTransitiveGroups}\left(34\,'\mathrm{output}'=''iterator''\right)$

$\mathrm{it}≔\mathrm{\⟨Transitive\; Groups\; Iterator:\; 34/1\; ..\; 34/115\⟩}$

$A≔\mathrm{Array}\left(\left[\right]\right)\:$

$$\begin{gathered} \mathbf{for}\mathrm{id},G\mathbf{in}\mathrm{it}\mathbf{do} \\ \mathbf{if}\mathrm{IsSemiRegular}\left(G\right)\mathbf{then} \\ \mathrm{`,=`}\left(A\,\left[\mathrm{id}\right]\right) \\ \mathbf{end}\mathbf{if} \\ \mathbf{end}\mathbf{do}\: \end{gathered}$$

$\left[\mathrm{seq}\right]\left(A\right)$

$\left[\left[34\,1\right]\,\left[34\,2\right]\right]$