IsRegular - Maple Help

GroupTheory

 IsRegular
 determine whether a permutation group is regular
 IsSemiRegular
 determine whether a permutation group is semi-regular

 Calling Sequence IsRegular( G, dom ) IsSemiRegular( G, dom )

Parameters

 G - : PermutationGroup : a permutation group dom - : {set,list}(posint) : (optional) a domain on which G acts

Description

 • 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].

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Group}\left(\left\{\mathrm{Perm}\left(\left[\left[1,2\right]\right]\right),\mathrm{Perm}\left(\left[\left[1,2,3\right],\left[4,5\right]\right]\right)\right\}\right)$
 ${G}{≔}⟨\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}\right){,}\left({1}{,}{2}\right)⟩$ (1)
 > $\mathrm{IsRegular}\left(G\right)$
 ${\mathrm{false}}$ (2)
 > $\mathrm{IsRegular}\left(\mathrm{CyclicGroup}\left(6\right)\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{IsRegular}\left(\mathrm{CyclicGroup}\left(6,':-\mathrm{mindegree}'\right)\right)$
 ${\mathrm{false}}$ (4)
 > $G≔\mathrm{Group}\left(\left\{\mathrm{Perm}\left(\left[\left[1,2\right],\left[3,4\right]\right]\right)\right\}\right)$
 ${G}{≔}⟨\left({1}{,}{2}\right)\left({3}{,}{4}\right)⟩$ (5)
 > $\mathrm{IsSemiRegular}\left(G\right)$
 ${\mathrm{true}}$ (6)
 > $\mathrm{IsRegular}\left(G\right)$
 ${\mathrm{false}}$ (7)
 > $G≔\mathrm{Group}\left(\left\{\mathrm{Perm}\left(\left[\left[1,2\right],\left[3,4,5\right]\right]\right)\right\}\right)$
 ${G}{≔}⟨\left({1}{,}{2}\right)\left({3}{,}{4}{,}{5}\right)⟩$ (8)
 > $\mathrm{IsSemiRegular}\left(G\right)$
 ${\mathrm{false}}$ (9)
 > $G≔\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[2,3,4\right]\right]\right),\mathrm{Perm}\left(\left[\left[2,3\right]\right]\right)\right]\right)$
 ${G}{≔}⟨\left({2}{,}{3}{,}{4}\right){,}\left({2}{,}{3}\right)⟩$ (10)
 > $\mathrm{IsSemiRegular}\left(G\right)$
 ${\mathrm{false}}$ (11)
 > $\mathrm{IsSemiRegular}\left(\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[2,3,4\right]\right]\right),\mathrm{Perm}\left(\left[\left[5,6,7\right]\right]\right)\right]\right)\right)$
 ${\mathrm{false}}$ (12)
 > $G≔\mathrm{Group}\left(\left\{\mathrm{Perm}\left(\left[\left[1,2,3\right],\left[4,5,6\right]\right]\right)\right\}\right)$
 ${G}{≔}⟨\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}{,}{6}\right)⟩$ (13)
 > $\mathrm{IsSemiRegular}\left(G\right)$
 ${\mathrm{true}}$ (14)
 > $G≔\mathrm{Group}\left(\left\{\mathrm{Perm}\left(\left[\left[1,2,3\right]\right]\right),\mathrm{Perm}\left(\left[\left[4,5,6\right]\right]\right)\right\}\right)$
 ${G}{≔}⟨\left({1}{,}{2}{,}{3}\right){,}\left({4}{,}{5}{,}{6}\right)⟩$ (15)
 > $\mathrm{IsSemiRegular}\left(G\right)$
 ${\mathrm{false}}$ (16)
 > $\mathrm{IsRegular}\left(G,\left\{1,2,3\right\}\right)$
 ${\mathrm{true}}$ (17)
 > $\mathrm{IsRegular}\left(G,\left\{4,5,6\right\}\right)$
 ${\mathrm{true}}$ (18)

The symmetric group in its natural permutation representation is not regular.

 > $\mathrm{IsRegular}\left(\mathrm{Symm}\left(3\right)\right)$
 ${\mathrm{false}}$ (19)

But there is a regular permutation representation of degree $6$ (the order of the group).

 > $\mathrm{IsRegular}\left(\mathrm{TransitiveGroup}\left(6,2\right)\right)$
 ${\mathrm{true}}$ (20)
 > $\mathrm{AreIsomorphic}\left(\mathrm{Symm}\left(3\right),\mathrm{TransitiveGroup}\left(6,2\right)\right)$
 ${\mathrm{true}}$ (21)

The quaternion group of order $8$ has a regular representation of degree $8$.

 > $\mathrm{AreIsomorphic}\left(\mathrm{TransitiveGroup}\left(8,5\right),\mathrm{QuaternionGroup}\left(\right)\right)$
 ${\mathrm{true}}$ (22)
 > $\mathrm{IsRegular}\left(\mathrm{TransitiveGroup}\left(8,5\right)\right)$
 ${\mathrm{true}}$ (23)

We construct a diagonal embedding into the direct square.

 > $G≔\mathrm{DirectProduct}\left(\mathrm{TransitiveGroup}\left(8,5\right),\mathrm{TransitiveGroup}\left(8,5\right)\right)$
 ${G}{≔}⟨\left({1}{,}{2}{,}{3}{,}{8}\right)\left({4}{,}{5}{,}{6}{,}{7}\right){,}\left({1}{,}{7}{,}{3}{,}{5}\right)\left({2}{,}{6}{,}{8}{,}{4}\right){,}\left({9}{,}{10}{,}{11}{,}{16}\right)\left({12}{,}{13}{,}{14}{,}{15}\right){,}\left({9}{,}{15}{,}{11}{,}{13}\right)\left({10}{,}{14}{,}{16}{,}{12}\right)⟩$ (24)
 > $\mathrm{IsSemiRegular}\left(G\right)$
 ${\mathrm{false}}$ (25)
 > $\mathrm{gens}≔\mathrm{Generators}\left(G\right):$

This is the diagonal subgroup.

 > $H≔\mathrm{Subgroup}\left(\left\{\mathrm{gens}\left[1\right]·\mathrm{gens}\left[3\right],\mathrm{gens}\left[2\right]·\mathrm{gens}\left[4\right]\right\},G\right)$
 ${H}{≔}⟨\left({1}{,}{2}{,}{3}{,}{8}\right)\left({4}{,}{5}{,}{6}{,}{7}\right)\left({9}{,}{10}{,}{11}{,}{16}\right)\left({12}{,}{13}{,}{14}{,}{15}\right){,}\left({1}{,}{7}{,}{3}{,}{5}\right)\left({2}{,}{6}{,}{8}{,}{4}\right)\left({9}{,}{15}{,}{11}{,}{13}\right)\left({10}{,}{14}{,}{16}{,}{12}\right)⟩$ (26)
 > $\mathrm{AreIsomorphic}\left(H,\mathrm{QuaternionGroup}\left(\right)\right)$
 ${\mathrm{true}}$ (27)
 > $\mathrm{IsSemiRegular}\left(H\right)$
 ${\mathrm{true}}$ (28)

Frobenius groups are never regular.

 > $\mathrm{IsRegular}\left(\mathrm{FrobeniusGroup}\left(15000,3\right)\right)$
 ${\mathrm{false}}$ (29)

Let's find all the regular groups of degree $34$. First, we create an iterator for all the transitive groups of that degree.

 > $\mathrm{it}≔\mathrm{AllTransitiveGroups}\left(34,'\mathrm{output}'="iterator"\right)$
 ${\mathrm{it}}{≔}{\mathrm{⟨Transitive Groups Iterator: 34/1 .. 34/115⟩}}$ (30)

Create an Array in which to store the transitive group IDs of those that are found to be regular.

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

Now iterate over the groups and check for regularity.  Since we already know that the groups are transitive, we avoid the redundant transitivity check and test only for semi-regularity.

 > $\mathbf{for}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{id},G\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{it}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathbf{if}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{IsSemiRegular}\left(G\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{then}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathrm{,=}\left(A,\left[\mathrm{id}\right]\right)\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathbf{end}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{if}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathbf{end}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}:$
 > $\left[\mathrm{seq}\right]\left(A\right)$
 $\left[\left[{34}{,}{1}\right]{,}\left[{34}{,}{2}\right]\right]$ (31)