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-centralising 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 $G$.

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

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

$\mathrm{true}$

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

$\mathrm{true}$

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

$⟨\left(1\,2\,3\right)⟩$

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

$⟨\left(1\,2\right)⟩$

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

$\mathrm{true}$

$G≔\mathrm{SmallGroup}\left(6\,1\right)$

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

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

$\mathrm{true}$

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

$\mathrm{true}$

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

$\mathrm{false}$

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

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

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

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

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

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

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

$\mathrm{true}$

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

$\mathrm{true}$

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

$\mathrm{false}$

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

$\mathrm{true}$

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

$\mathrm{false}$

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

$\mathrm{true}$

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

$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)$

$110$

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

$\mathrm{true}$

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

$\mathrm{false}$

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

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

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

$\mathrm{true}$

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

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

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

$11$

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

$C≔\mathrm{<\; a\; permutation\; group\; on\; 11\; letters\; with\; 6\; generators\; >}$

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

$10$

$\mathrm{IsMalnormal}\left(C\&comma;P\right)$

$\mathrm{true}$

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

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

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

$11$

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

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

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

$10$

$\mathrm{IsMalnormal}\left(C\&comma;G\right)$

$\mathrm{true}$

$G≔\mathrm{DihedralGroup}\left(7\right)\&colon;$

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

$C≔⟨\left(1\&comma;4\right)\left(2\&comma;3\right)\left(5\&comma;7\right)⟩$

$\mathrm{IsMalnormal}\left(C\&comma;G\right)$

$\mathrm{true}$

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

$G≔M_{10}$

$S≔\mathrm{Stabiliser}\left(1\&comma;G\right)$

$S≔\mathrm{<\; a\; permutation\; group\; on\; 10\; letters\; with\; 7\; generators\; >}$

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

$72$

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

$\mathrm{true}$

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

$\mathrm{true}$

$\mathrm{AreIsomorphic}\left(\mathrm{FrobeniusComplement}\left(S\right)\&comma;\mathrm{QuaternionGroup}\left(\right)\right)$

$\mathrm{true}$

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.