A permutation group $G$ is a Frobenius group if it is transitive, has a non-trivial point stabilizer, and no non-trivial element of $G$ fixes more than one point.

The IsFrobeniusPermGroup( G ) command returns true if the permutation group G is a Frobenius group, and returns false otherwise.

An abstract group $G$ is a Frobenius group if it has a proper, non-trivial malnormal subgroup self-centralizing subgroup $H$, called a Frobenius complement. In this case, $G$ has a normal (even characteristic) subgroup $K$, called the Frobenius kernel, consisting of the identity element of $G$ and the elements of $G$ that do not belong to any conjugate of $H$ in $G$.

The IsFrobeniusGroup( G ) command returns true if G is a Frobenius group as an abstract group, and returns false otherwise.

The two definitions are equivalent in the following sense.  If $G$ is a Frobenius permutation group, then $G$ is Frobenius as an abstract group, with the stabilizer of a point being a Frobenius complement in $G$. Conversely, if $G$ is Frobenius as an abstract group, then the action of $G$ on the cosets of a Frobenius complement is faithful and is Frobenius as a permutation group, and so $G$ is isomorphic to the corresponding Frobenius permutation group,

The Frobenius kernel of a Frobenius group $G$ is uniquely defined, because a group can be a Frobenius group in at most one way. The Frobenius complement of a Frobenius group $G$ is well-defined up to conjugacy in $G$.

If $G$ is a Frobenius group, the FrobeniusKernel( G ) command returns the Frobenius kernel of $G$.  If $G$ is not Frobenius, an exception is raised.

If $G$ is a Frobenius group, the FrobeniusComplement( G ) command returns a Frobenius complement of $G$.  If $G$ is not Frobenius, an exception is raised.

For a Frobenius group G, the FrobeniusPermRep( G ) command returns a Frobenius permutation group isomorphic to $G$. It is permutation isomorphic to the action on $G$ on the cosets of a Frobenius complement in $G$.

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

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

$\mathrm{true}$

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

$\mathrm{true}$

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

Warning, over-writing property `["FrobeniusKernel"]' with a different value

$⟨\left(1\,2\,3\right)⟩$

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

Warning, over-writing property `["FrobeniusComplement"]' with a different value

$⟨\left(1\,2\right)⟩$

$\mathrm{IsMalnormal}\left(\mathrm{FrobeniusComplement}\left(\mathrm{Symm}\left(3\right)\right)\,\mathrm{Symm}\left(3\right)\right)$

Warning, over-writing property `["FrobeniusComplement"]' with a different value

$\mathrm{true}$

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

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

$\mathrm{AreIsomorphic}\left(G\,\mathrm{Symm}\left(3\right)\right)$

$\mathrm{true}$

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

$\mathrm{true}$

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

$\mathrm{false}$

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

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

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

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

$H≔\mathrm{FrobeniusPermRep}\left(G\right)$

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

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

$\mathrm{true}$

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

$\mathrm{true}$

$\mathrm{IsFrobeniusGroup}\left(\mathrm{DihedralGroup}\left(4\right)\right)$

$\mathrm{false}$

$\mathrm{IsFrobeniusGroup}\left(\mathrm{DihedralGroup}\left(5\right)\right)$

$\mathrm{true}$

$\mathrm{IsFrobeniusGroup}\left(\mathrm{DihedralGroup}\left(6\right)\right)$

$\mathrm{false}$

$\mathrm{IsFrobeniusGroup}\left(\mathrm{PSL}\left(2\,3\right)\right)$

$\mathrm{true}$

$\mathrm{IsFrobeniusPermGroup}\left(\mathrm{AGL}\left(1\,243\right)\right)$

$\mathrm{true}$

$a,b≔\mathrm{op}\left(\mathrm{Generators}\left(\mathrm{JankoGroup}\left(1\right)\right)\right)\:$

$u≔{\left(a·b\right)}^{-2}·b·a·b\:$$\mathrm{PermOrder}\left(u\right)$

$2$

$v≔{\left(a·b·b\right)}^{-2}·{\left(a·b·a·b·a·b·b·a·b·a·b·b\right)}^3·{\left(a·b·b\right)}^2\:$$\mathrm{PermOrder}\left(v\right)$

$5$

$G≔\mathrm{Group}\left(\left[u\,v\right]\right)\:$$\mathrm{GroupOrder}\left(G\right)$

$110$

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

$\mathrm{true}$

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

$\mathrm{false}$

$P≔\mathrm{FrobeniusPermRep}\left(G\right)$

$P≔⟨\left(1\,8\right)\left(2\,4\right)\left(3\,7\right)\left(6\,10\right)\left(9\,11\right)\,\left(1\,9\,7\,11\,10\right)\left(2\,5\,3\,8\,4\right)⟩$

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

$\mathrm{true}$

$K≔\mathrm{FrobeniusKernel}\left(P\right)$

$K≔⟨\left(1\,7\,9\,4\,10\,6\,2\,11\,3\,8\,5\right)⟩$

$\mathrm{GroupOrder}\left(K\right)$

$11$

$C≔\mathrm{FrobeniusComplement}\left(P\right)$

$C≔⟨\left(2\,8\,3\,7\,11\,6\,9\,4\,5\,10\right)⟩$

$\mathrm{GroupOrder}\left(C\right)$

$10$

$\mathrm{IsMalnormal}\left(C\,P\right)$

$\mathrm{true}$

$K≔\mathrm{FrobeniusKernel}\left(G\right)$

$K≔Fitt\left(⟨\left(1\,255\right)\left(2\,260\right)\left(3\,181\right)\left(4\,247\right)\left(5\,53\right)\left(6\,88\right)\left(7\,218\right)\left(8\,183\right)\left(9\,221\right)\left(10\,17\right)\left(11\,67\right)\left(12\,133\right)\left(13\,50\right)\left(14\,239\right)\left(15\,39\right)\left(16\,43\right)\left(18\,241\right)\left(19\,61\right)\left(20\,202\right)\left(21\,85\right)\left(22\,262\right)\left(23\,227\right)\left(24\,132\right)\left(25\,86\right)\left(26\,217\right)\left(27\,261\right)\left(28\,123\right)\left(29\,77\right)\left(30\,124\right)\left(31\,149\right)\left(32\,102\right)\left(33\,184\right)\left(34\,153\right)\left(35\,83\right)\left(36\,125\right)\left(37\,238\right)\left(38\,209\right)\left(40\,168\right)\left(41\,249\right)\left(42\,215\right)\left(44\,164\right)\left(45\,167\right)\left(46\,145\right)\left(48\,232\right)\left(49\,194\right)\left(51\,170\right)\left(52\,62\right)\left(54\,74\right)\left(55\,138\right)\left(56\,199\right)\left(57\,236\right)\left(58\,203\right)\left(59\,231\right)\left(60\,69\right)\left(63\,266\right)\left(64\,165\right)\left(65\,254\right)\left(68\,201\right)\left(70\,169\right)\left(71\,148\right)\left(72\,79\right)\left(73\,198\right)\left(75\,226\right)\left(76\,173\right)\left(78\,166\right)\left(80\,150\right)\left(81\,96\right)\left(82\,193\right)\left(84\,99\right)\left(89\,107\right)\left(90\,110\right)\left(91\,163\right)\left(92\,206\right)\left(93\,143\right)\left(94\,178\right)\left(95\,237\right)\left(97\,245\right)\left(98\,174\right)\left(100\,242\right)\left(101\,126\right)\left(103\,187\right)\left(104\,189\right)\left(105\,157\right)\left(106\,185\right)\left(109\,155\right)\left(111\,213\right)\left(112\,195\right)\left(114\,176\right)\left(115\,191\right)\left(116\,171\right)\left(117\,252\right)\left(118\,250\right)\left(119\,253\right)\left(120\,136\right)\left(121\,161\right)\left(122\,140\right)\left(127\,265\right)\left(128\,208\right)\left(129\,147\right)\left(131\,223\right)\left(134\,200\right)\left(135\,152\right)\left(137\,224\right)\left(139\,192\right)\left(141\,207\right)\left(142\,258\right)\left(144\,244\right)\left(146\,159\right)\left(151\,158\right)\left(154\,233\right)\left(156\,162\right)\left(160\,196\right)\left(172\,263\right)\left(175\,228\right)\left(177\,222\right)\left(179\,230\right)\left(180\,220\right)\left(182\,204\right)\left(190\,259\right)\left(197\,225\right)\left(205\,256\right)\left(210\,240\right)\left(211\,212\right)\left(216\,257\right)\left(219\,248\right)\left(229\,264\right)\left(234\,246\right)\left(243\,251\right)\,\left(1\,26\,257\,192\,241\right)\left(2\,83\,3\,119\,180\right)\left(4\,94\,63\,13\,217\right)\left(5\,64\,196\,193\,106\right)\left(6\,211\,40\,136\,82\right)\left(7\,126\,264\,141\,209\right)\left(8\,244\,163\,59\,243\right)\left(9\,142\,16\,232\,109\right)\left(10\,88\,99\,165\,111\right)\left(11\,70\,250\,265\,87\right)\left(12\,123\,226\,134\,66\right)\left(14\,21\,29\,216\,49\right)\left(15\,201\,219\,221\,255\right)\left(17\,28\,53\,179\,121\right)\left(18\,44\,43\,190\,175\right)\left(19\,159\,122\,157\,108\right)\left(20\,150\,45\,81\,54\right)\left(22\,48\,234\,57\,107\right)\left(23\,154\,224\,252\,58\right)\left(24\,61\,168\,213\,240\right)\left(25\,100\,67\,253\,261\right)\left(27\,80\,35\,95\,30\right)\left(31\,186\,74\,263\,127\right)\left(32\,156\,76\,223\,247\right)\left(33\,68\,246\,207\,50\right)\left(34\,77\,62\,178\,236\right)\left(36\,259\,194\,197\,38\right)\left(37\,195\,102\,152\,258\right)\left(39\,71\,42\,135\,85\right)\left(41\,153\,177\,148\,78\right)\left(46\,72\,131\,176\,139\right)\left(47\,151\,167\,98\,97\right)\left(51\,181\,245\,118\,214\right)\left(52\,155\,162\,65\,125\right)\left(55\,91\,189\,144\,171\right)\left(56\,104\,191\,90\,116\right)\left(60\,75\,130\,120\,210\right)\left(69\,235\,230\,160\,105\right)\left(73\,228\,249\,79\,262\right)\left(84\,146\,161\,133\,132\right)\left(86\,149\,174\,113\,220\right)\left(89\,238\,114\,229\,239\right)\left(93\,227\,205\,117\,206\right)\left(96\,124\,172\,260\,170\right)\left(101\,248\,145\,254\,222\right)\left(103\,203\,137\,128\,204\right)\left(110\,251\,231\,183\,199\right)\left(112\,166\,266\,164\,218\right)\left(140\,188\,185\,212\,200\right)\left(143\,182\,208\,187\,256\right)\left(158\,242\,237\,169\,202\right)\left(173\,184\,198\,215\,225\right)⟩\right)$

$\mathrm{GroupOrder}\left(K\right)$

$11$

$C≔\mathrm{FrobeniusComplement}\left(G\right)$

$C≔⟨\left(1\,153\right)\left(3\,98\right)\left(4\,71\right)\left(6\,108\right)\left(7\,223\right)\left(8\,55\right)\left(9\,63\right)\left(10\,210\right)\left(11\,174\right)\left(12\,157\right)\left(13\,79\right)\left(14\,249\right)\left(15\,218\right)\left(16\,29\right)\left(17\,188\right)\left(18\,135\right)\left(19\,75\right)\left(21\,50\right)\left(22\,39\right)\left(23\,117\right)\left(24\,106\right)\left(25\,127\right)\left(26\,37\right)\left(27\,263\right)\left(28\,60\right)\left(30\,70\right)\left(31\,51\right)\left(32\,49\right)\left(33\,43\right)\left(34\,209\right)\left(35\,167\right)\left(36\,217\right)\left(38\,258\right)\left(40\,179\right)\left(41\,156\right)\left(42\,264\right)\left(44\,254\right)\left(45\,149\right)\left(46\,197\right)\left(47\,261\right)\left(48\,247\right)\left(52\,112\right)\left(53\,61\right)\left(54\,119\right)\left(56\,189\right)\left(57\,145\right)\left(58\,227\right)\left(59\,91\right)\left(62\,201\right)\left(64\,130\right)\left(65\,85\right)\left(66\,213\right)\left(67\,81\right)\left(68\,176\right)\left(69\,185\right)\left(72\,142\right)\left(73\,195\right)\left(74\,97\right)\left(76\,255\right)\left(77\,131\right)\left(78\,246\right)\left(80\,86\right)\left(83\,87\right)\left(84\,235\right)\left(88\,123\right)\left(89\,173\right)\left(90\,251\right)\left(92\,137\right)\left(93\,182\right)\left(94\,229\right)\left(95\,181\right)\left(99\,212\right)\left(100\,170\right)\left(101\,216\right)\left(102\,248\right)\left(103\,154\right)\left(104\,171\right)\left(105\,240\right)\left(107\,164\right)\left(109\,215\right)\left(110\,138\right)\left(111\,160\right)\left(113\,124\right)\left(114\,190\right)\left(115\,163\right)\left(116\,144\right)\left(118\,220\right)\left(120\,140\right)\left(122\,196\right)\left(125\,262\right)\left(126\,232\right)\left(128\,224\right)\left(129\,147\right)\left(133\,193\right)\left(134\,230\right)\left(136\,161\right)\left(139\,166\right)\left(141\,241\right)\left(143\,233\right)\left(146\,200\right)\left(148\,259\right)\left(150\,214\right)\left(151\,265\right)\left(152\,236\right)\left(155\,192\right)\left(158\,260\right)\left(159\,165\right)\left(162\,207\right)\left(169\,245\right)\left(172\,250\right)\left(175\,178\right)\left(177\,238\right)\left(180\,202\right)\left(183\,191\right)\left(184\,222\right)\left(186\,237\right)\left(187\,206\right)\left(194\,234\right)\left(198\,257\right)\left(199\,243\right)\left(203\,208\right)\left(204\,256\right)\left(205\,252\right)\left(211\,226\right)\left(219\,228\right)\left(221\,239\right)\left(225\,266\right)\left(231\,244\right)\,\left(1\,207\,166\,62\,223\right)\left(2\,20\,253\,96\,242\right)\left(3\,87\,158\,220\,30\right)\left(4\,125\,21\,41\,234\right)\left(5\,132\,82\,121\,168\right)\left(6\,193\,140\,159\,226\right)\left(7\,153\,162\,139\,201\right)\left(8\,56\,244\,138\,90\right)\left(9\,42\,247\,218\,89\right)\left(10\,196\,130\,61\,185\right)\left(11\,35\,25\,47\,54\right)\left(12\,111\,146\,235\,106\right)\left(13\,221\,175\,176\,29\right)\left(14\,102\,198\,177\,255\right)\left(15\,173\,63\,264\,48\right)\left(16\,79\,239\,178\,68\right)\left(17\,161\,134\,212\,60\right)\left(18\,94\,148\,37\,209\right)\left(19\,88\,240\,66\,179\right)\left(22\,44\,152\,219\,52\right)\left(24\,157\,160\,200\,84\right)\left(26\,34\,135\,229\,259\right)\left(27\,169\,186\,170\,151\right)\left(28\,188\,136\,230\,99\right)\left(31\,86\,250\,124\,81\right)\left(32\,232\,77\,38\,145\right)\left(33\,85\,241\,109\,101\right)\left(36\,142\,266\,192\,73\right)\left(39\,254\,236\,228\,112\right)\left(40\,75\,123\,105\,213\right)\left(43\,65\,141\,215\,216\right)\left(45\,214\,180\,74\,95\right)\left(46\,78\,190\,184\,107\right)\left(49\,126\,131\,258\,57\right)\left(50\,156\,194\,71\,262\right)\left(51\,80\,172\,113\,67\right)\left(53\,69\,210\,122\,64\right)\left(55\,189\,231\,110\,251\right)\left(58\,256\,92\,224\,187\right)\left(59\,191\,116\,243\,163\right)\left(70\,98\,83\,260\,118\right)\left(72\,225\,155\,195\,217\right)\left(76\,249\,248\,257\,238\right)\left(91\,183\,144\,199\,115\right)\left(93\,103\,205\,203\,233\right)\left(97\,181\,149\,150\,202\right)\left(100\,265\,263\,245\,237\right)\left(108\,133\,120\,165\,211\right)\left(114\,222\,164\,197\,246\right)\left(119\,174\,167\,127\,261\right)\left(128\,206\,227\,204\,137\right)\left(143\,182\,154\,252\,208\right)⟩$

$\mathrm{GroupOrder}\left(C\right)$

$10$

$\mathrm{IsMalnormal}\left(C\,G\right)$

$\mathrm{true}$

$G≔\mathrm{DihedralGroup}\left(7\right)\:$

$C≔\mathrm{FrobeniusComplement}\left(G\right)$

Warning, over-writing property `["FrobeniusComplement"]' with a different value

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

$\mathrm{IsMalnormal}\left(C\,G\right)$

$\mathrm{true}$

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

$G≔M_{10}$

$S≔\mathrm{Stabilizer}\left(1\,G\right)$

$S≔⟨\left(2\,6\,10\,7\right)\left(3\,9\,4\,5\right)\,\left(2\,6\,8\,3\right)\left(4\,9\,5\,7\right)⟩$

$\mathrm{GroupOrder}\left(S\right)$

$72$

$\mathrm{IsFrobeniusGroup}\left(S\right)$

$\mathrm{true}$

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

$\mathrm{true}$

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

$\mathrm{true}$

The GroupTheory[IsFrobeniusPermGroup], GroupTheory[IsFrobeniusGroup], GroupTheory[FrobeniusKernel], GroupTheory[FrobeniusComplement] and GroupTheory[FrobeniusPermRep] commands were introduced in Maple 2019.

For more information on Maple 2019 changes, see Updates in Maple 2019.