For a permutation group $G$, an "EARNS" is a normal subgroup of $G$ that is elementary abelian and acts regularly on the domain of action of $G$. A permutation group may, or may not, possess an EARNS.

The EARNS( G ) command returns an EARNS for a permutation group G, provided that one exists, and returns FAIL if G has no EARNS.

It is clear that for a permutation group to posess an EARNS it must be transitive and its support must have prime power cardinality. Therefore, EARNS returns FAIL if either of these conditions is not true.

In general, for Maple to identify an EARNS for a permutation group the group must either be primitive or a Frobenius group (or both). If G is neither primitive nor a Frobenius group, then EARNS may raise an exception indicating that the group is imprimitive and that Maple cannot, in that case, determine whether or not G has an EARNS.

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

$G≔\mathrm{Symm}\left(3\right)$

$G≔{\mathbf{S}}_3$

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

$⟨\left(1\,2\,3\right)⟩$

$\mathrm{EARNS}\left(\mathrm{Alt}\left(4\right)\right)$

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

$G≔\mathrm{Symm}\left(4\right)$

$G≔{\mathbf{S}}_4$

$E≔\mathrm{EARNS}\left(G\right)$

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

$\mathrm{IsNormal}\left(E\,G\right)$

$\mathrm{true}$

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

$\mathrm{true}$

$\mathrm{IsElementary}\left(E\right)$

$\mathrm{true}$

$G≔\mathrm{CyclicGroup}\left(10\right)$

$G≔C_{10}$

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

$\mathrm{FAIL}$

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

$\mathrm{true}$

$\mathrm{type}\left(\mathrm{SupportLength}\left(G\right)\,'\mathrm{primepower}'\right)$

$\mathrm{false}$

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

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

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

$\mathrm{false}$

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

$\mathrm{FAIL}$

$G≔\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[2\,7\,4\,8\,6\,5\,3\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[2\,4\,3\right]\,\left[6\,8\,7\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[1\,2\right]\,\left[3\,4\right]\,\left[5\,6\right]\,\left[7\,8\right]\right]\right)\right]\right)$

$G≔⟨\left(2\,7\,4\,8\,6\,5\,3\right)\,\left(2\,4\,3\right)\left(6\,8\,7\right)\,\left(1\,2\right)\left(3\,4\right)\left(5\,6\right)\left(7\,8\right)⟩$

$E≔\mathrm{EARNS}\left(G\right)$

$E≔⟨\left(1\,4\right)\left(2\,3\right)\left(5\,8\right)\left(6\,7\right)\,\left(1\,7\right)\left(2\,8\right)\left(3\,5\right)\left(4\,6\right)\,\left(1\,2\right)\left(3\,4\right)\left(5\,6\right)\left(7\,8\right)⟩$

$G≔\mathrm{FrobeniusGroup}\left(14520\,2\right)$

$G≔⟨\left(2\,6\,74\,80\,16\right)\left(3\,49\,63\,116\,19\right)\left(4\,51\,108\,68\,59\right)\left(5\,53\,52\,110\,40\right)\left(7\,10\,37\,47\,78\right)\left(8\,17\,61\,60\,90\right)\left(9\,20\,89\,97\,93\right)\left(11\,43\,106\,48\,31\right)\left(12\,103\,70\,29\,36\right)\left(13\,104\,85\,86\,84\right)\left(14\,81\,38\,42\,41\right)\left(15\,58\,64\,79\,77\right)\left(18\,105\,56\,101\,91\right)\left(21\,102\,109\,54\,112\right)\left(22\,26\,100\,55\,32\right)\left(23\,39\,28\,33\,35\right)\left(24\,83\,98\,71\,75\right)\left(25\,118\,76\,119\,95\right)\left(27\,30\,117\,94\,92\right)\left(34\,99\,120\,69\,72\right)\left(44\,62\,113\,87\,82\right)\left(45\,121\,88\,66\,57\right)\left(46\,115\,107\,114\,65\right)\left(50\,111\,67\,96\,73\right)\,\left(2\,22\,21\,99\,104\right)\left(3\,58\,75\,73\,20\right)\left(4\,26\,19\,65\,78\right)\left(5\,64\,16\,92\,36\right)\left(6\,9\,91\,37\,17\right)\left(7\,94\,85\,96\,106\right)\left(8\,52\,54\,57\,63\right)\left(10\,23\,102\,15\,38\right)\left(11\,95\,93\,87\,109\right)\left(12\,114\,89\,34\,39\right)\left(13\,119\,70\,81\,49\right)\left(14\,108\,98\,113\,74\right)\left(18\,84\,45\,83\,35\right)\left(24\,32\,61\,103\,43\right)\left(25\,33\,68\,90\,30\right)\left(27\,116\,62\,69\,56\right)\left(28\,67\,44\,100\,40\right)\left(29\,101\,50\,112\,51\right)\left(31\,110\,41\,115\,105\right)\left(42\,55\,121\,97\,117\right)\left(46\,80\,66\,111\,118\right)\left(47\,76\,120\,71\,53\right)\left(48\,72\,88\,79\,59\right)\left(60\,77\,82\,86\,107\right)\,\left(2\,3\right)\left(4\,5\right)\left(6\,49\right)\left(7\,12\right)\left(8\,14\right)\left(9\,13\right)\left(10\,103\right)\left(11\,35\right)\left(15\,32\right)\left(16\,19\right)\left(17\,81\right)\left(18\,95\right)\left(20\,104\right)\left(21\,75\right)\left(22\,58\right)\left(23\,43\right)\left(24\,102\right)\left(25\,105\right)\left(26\,64\right)\left(27\,46\right)\left(28\,48\right)\left(29\,47\right)\left(30\,115\right)\left(31\,33\right)\left(34\,96\right)\left(36\,78\right)\left(37\,70\right)\left(38\,61\right)\left(39\,106\right)\left(40\,59\right)\left(41\,90\right)\left(42\,60\right)\left(44\,88\right)\left(45\,87\right)\left(50\,120\right)\left(51\,53\right)\left(52\,108\right)\left(54\,98\right)\left(55\,77\right)\left(56\,118\right)\left(57\,113\right)\left(62\,66\right)\left(63\,74\right)\left(65\,92\right)\left(67\,72\right)\left(68\,110\right)\left(69\,111\right)\left(71\,112\right)\left(73\,99\right)\left(76\,101\right)\left(79\,100\right)\left(80\,116\right)\left(82\,121\right)\left(83\,109\right)\left(84\,93\right)\left(85\,89\right)\left(86\,97\right)\left(91\,119\right)\left(94\,114\right)\left(107\,117\right)\,\left(1\,2\,7\,27\,79\,53\,51\,100\,46\,12\,3\right)\left(4\,8\,28\,72\,52\,84\,70\,112\,102\,47\,13\right)\left(5\,9\,29\,24\,71\,37\,93\,108\,67\,48\,14\right)\left(6\,22\,69\,97\,66\,20\,17\,15\,19\,35\,23\right)\left(10\,38\,94\,107\,50\,36\,33\,87\,76\,82\,39\right)\left(11\,16\,32\,81\,104\,62\,86\,111\,58\,49\,43\right)\left(18\,59\,75\,110\,105\,99\,44\,26\,77\,113\,60\right)\left(21\,40\,95\,42\,57\,55\,64\,88\,73\,25\,68\right)\left(30\,85\,65\,116\,118\,83\,34\,90\,98\,119\,74\right)\left(31\,78\,120\,117\,114\,61\,103\,106\,121\,101\,45\right)\left(41\,96\,109\,56\,80\,92\,89\,115\,63\,91\,54\right)\,\left(1\,4\,16\,26\,78\,65\,92\,36\,64\,19\,5\right)\left(2\,8\,32\,77\,120\,116\,89\,33\,88\,35\,9\right)\left(3\,13\,11\,44\,31\,85\,80\,50\,55\,15\,14\right)\left(6\,24\,27\,72\,104\,60\,114\,83\,63\,76\,25\right)\left(7\,28\,81\,113\,117\,118\,115\,87\,73\,23\,29\right)\left(10\,40\,97\,93\,51\,70\,111\,75\,106\,98\,41\right)\left(12\,47\,43\,99\,45\,30\,56\,107\,57\,17\,48\right)\left(18\,61\,34\,91\,82\,68\,22\,71\,79\,52\,62\right)\left(20\,67\,46\,102\,49\,105\,101\,74\,109\,94\,42\right)\left(21\,69\,37\,53\,84\,86\,59\,103\,90\,54\,39\right)\left(38\,95\,66\,108\,100\,112\,58\,110\,121\,119\,96\right)⟩$

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

$\mathrm{true}$

$E≔\mathrm{EARNS}\left(G\right)\:$

$\mathrm{AreIsomorphic}\left(E\,\mathrm{ElementaryGroup}\left(11\,2\right)\right)$

$\mathrm{true}$

$\mathrm{IsSubgroup}\left(E\,\mathrm{FrobeniusKernel}\left(G\right)\right)\mathbf{and}\mathrm{IsSubgroup}\left(\mathrm{FrobeniusKernel}\left(G\right)\,E\right)$

$\mathrm{true}$

$G≔\mathrm{Symm}\left(2048\right)$

$G≔{\mathbf{S}}_{2048}$

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

$\mathrm{FAIL}$

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

$\mathrm{true}$

$G≔\mathrm{Alt}\left(5^5\right)$

$G≔{\mathbf{A}}_{3125}$

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

$\mathrm{FAIL}$

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

$\mathrm{true}$

$G≔\mathrm{TransitiveGroup}\left(27\,4\right)\:$

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

$⟨\left(1\,2\,3\right)\left(4\,5\,6\right)\left(7\,8\,9\right)\left(10\,11\,12\right)\left(13\,14\,15\right)\left(16\,17\,18\right)\left(19\,20\,21\right)\left(22\,23\,24\right)\left(25\,26\,27\right)\,\left(1\,6\,26\right)\left(2\,4\,27\right)\left(3\,5\,25\right)\left(7\,10\,13\right)\left(8\,11\,14\right)\left(9\,12\,15\right)\left(16\,21\,23\right)\left(17\,19\,24\right)\left(18\,20\,22\right)\,\left(1\,11\,19\right)\left(2\,12\,20\right)\left(3\,10\,21\right)\left(4\,15\,22\right)\left(5\,13\,23\right)\left(6\,14\,24\right)\left(7\,16\,25\right)\left(8\,17\,26\right)\left(9\,18\,27\right)⟩$

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

$\mathrm{false}$

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

$\mathrm{true}$

$G≔\mathrm{TransitiveGroup}\left(25\,9\right)$

$G≔⟨\left(2\,3\,5\,4\right)\left(6\,14\,24\,17\right)\left(7\,11\,23\,20\right)\left(8\,13\,22\,18\right)\left(9\,15\,21\,16\right)\left(10\,12\,25\,19\right)\,\left(1\,7\,11\,20\,23\right)\left(2\,8\,12\,16\,24\right)\left(3\,9\,13\,17\,25\right)\left(4\,10\,14\,18\,21\right)\left(5\,6\,15\,19\,22\right)\,\left(1\,25\,19\,12\,10\right)\left(2\,21\,20\,13\,6\right)\left(3\,22\,16\,14\,7\right)\left(4\,23\,17\,15\,8\right)\left(5\,24\,18\,11\,9\right)⟩$

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

$Fitt\left(⟨\left(2\,3\,5\,4\right)\left(6\,14\,24\,17\right)\left(7\,11\,23\,20\right)\left(8\,13\,22\,18\right)\left(9\,15\,21\,16\right)\left(10\,12\,25\,19\right)\,\left(1\,7\,11\,20\,23\right)\left(2\,8\,12\,16\,24\right)\left(3\,9\,13\,17\,25\right)\left(4\,10\,14\,18\,21\right)\left(5\,6\,15\,19\,22\right)\,\left(1\,25\,19\,12\,10\right)\left(2\,21\,20\,13\,6\right)\left(3\,22\,16\,14\,7\right)\left(4\,23\,17\,15\,8\right)\left(5\,24\,18\,11\,9\right)⟩\right)$

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

$\mathrm{false}$

$G≔\mathrm{TransitiveGroup}\left(16\,63\right)$

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

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

$\mathrm{false}$

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

$\mathrm{true}$

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

$\mathrm{FAIL}$

$G≔\mathrm{WreathProduct}\left(\mathrm{CyclicGroup}\left(3\right)\,\mathrm{CyclicGroup}\left(3\right)\right)$

$G≔⟨\left(1\,2\,3\right)\,\left(1\,4\,7\right)\left(2\,5\,8\right)\left(3\,6\,9\right)⟩$

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

$\mathrm{false}$

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

Error, (in GroupTheory:-EARNS) group must be primitive