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 $G$.

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

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

$\mathrm{true}$

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

$\mathrm{true}$

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

$⟨\left(1\,2\,3\right)⟩$

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

$⟨\left(1\,2\right)⟩$

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

$\mathrm{true}$

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

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

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

$\mathrm{true}$

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

$\mathrm{true}$

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

$\mathrm{false}$

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

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

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

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

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

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

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

$\mathrm{true}$

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

$\mathrm{true}$

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

$\mathrm{false}$

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

$\mathrm{true}$

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

$\mathrm{false}$

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

$\mathrm{true}$

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

$\mathrm{true}$

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

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

$2$

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

$5$

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

$110$

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

$\mathrm{true}$

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

$\mathrm{false}$

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

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

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

$\mathrm{true}$

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

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

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

$11$

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

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

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

$10$

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

$\mathrm{true}$

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

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

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

$11$

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

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

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

$10$

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

$\mathrm{true}$

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

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

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

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

$\mathrm{true}$

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

$G≔M_{10}$

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

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

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

$72$

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

$\mathrm{true}$

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

$\mathrm{true}$

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

$\mathrm{true}$

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.