GroupTheory
IsFrobeniusPermGroup
determine whether a group is a Frobenius permutation group
IsFrobeniusGroup
determine whether a group is a Frobenius group
FrobeniusKernel
compute the Frobenius kernel of a Frobenius group
FrobeniusComplement
compute the Frobenius complement of a Frobenius group
FrobeniusPermRep
compute a Frobenius permutation group isomorphic to a given Frobenius group
Calling Sequence
Parameters
Description
Examples
Compatibility
IsFrobeniusPermGroup( G )
IsFrobeniusGroup( G )
FrobeniusKernel( G )
FrobeniusComplement( G )
FrobeniusPermRep( G )
G
-
a permutation group
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.
with⁡GroupTheory:
The smallest Frobenius group is the symmetric group of degree 3.
IsFrobeniusGroup⁡Symm⁡3
true
IsFrobeniusPermGroup⁡Symm⁡3
FrobeniusKernel⁡Symm⁡3
1,2,3
FrobeniusComplement⁡Symm⁡3
1,2
IsMalnormal⁡FrobeniusComplement⁡Symm⁡3,Symm⁡3
A different permutation group isomorphic to the symmetric group of degree 3 is a Frobenius group, but is not Frobenius as a permutation group.
G ≔ Group⁡Perm⁡1,2,3,6,4,5,Perm⁡1,3,4,2,5,6
G≔1,23,64,5,1,3,42,5,6
AreIsomorphic⁡G,Symm⁡3
IsFrobeniusGroup⁡G
IsFrobeniusPermGroup⁡G
false
FrobeniusKernel⁡G
1,3,42,5,6
FrobeniusComplement⁡G
1,62,43,5
H ≔ FrobeniusPermRep⁡G
H≔1,2,1,3,2
IsFrobeniusPermGroup⁡H
AreIsomorphic⁡H,G
The dihedral group Dn is Frobenius if, and only, if, n is odd.
IsFrobeniusGroup⁡DihedralGroup⁡4
IsFrobeniusGroup⁡DihedralGroup⁡5
IsFrobeniusGroup⁡DihedralGroup⁡6
IsFrobeniusGroup⁡PSL⁡2,3
IsFrobeniusPermGroup⁡AGL⁡1,243
We construct here a Frobenius subgroup of order 110 in the first Janko group.
a,b ≔ op⁡Generators⁡JankoGroup⁡1:
u ≔ `.`⁡`.`⁡a,b−2,b,a,b:PermOrder⁡u
2
v ≔ `.`⁡`.`⁡a,b,b−2,`.`⁡a,b,a,b,a,b,b,a,b,a,b,b3,`.`⁡a,b,b2:PermOrder⁡v
5
G ≔ Group⁡u,v:GroupOrder⁡G
110
However, this is not a Frobenius action; to get a Frobenius permutation group, use FrobeniusPermRep.
P ≔ FrobeniusPermRep⁡G
P≔1,72,113,65,98,10,1,11,8,5,102,3,6,4,9
IsFrobeniusPermGroup⁡P
Now we can compute the Frobenius kernel and complement, and determine their orders.
K ≔ FrobeniusKernel⁡P
K≔1,9,5,7,2,8,6,4,3,10,11
GroupOrder⁡K
11
C ≔ FrobeniusComplement⁡P
C≔2,9,7,10,83,5,6,4,11,2,5,9,6,7,4,10,11,8,3
GroupOrder⁡C
10
IsMalnormal⁡C,P
Of course, we obtain the same result by computing the Frobenius kernel and complement of G itself.
K ≔ FrobeniusKernel⁡G
K≔1,76,89,164,254,85,50,79,142,38,342,158,265,25,80,214,150,86,127,151,2603,95,186,237,181,98,11,81,96,67,1744,48,126,216,198,195,52,201,176,190,1485,61,75,211,99,235,84,212,226,19,536,123,60,185,24,132,106,69,28,88,1087,15,22,125,217,37,177,184,43,29,1318,191,163,59,244,116,104,56,243,90,1389,239,249,156,207,241,135,236,145,197,26610,188,17,210,105,12,193,82,133,157,24013,21,65,44,107,173,255,153,209,258,7214,221,63,225,46,57,152,18,141,162,4116,33,222,238,26,36,262,39,218,223,7720,180,118,172,27,47,261,263,250,220,20223,58,208,224,92,187,154,143,182,256,25230,113,124,70,245,74,119,253,54,97,16931,45,35,87,83,167,149,51,100,242,17032,248,228,178,229,42,109,192,166,246,19440,168,179,134,146,165,130,64,159,200,23049,234,78,139,155,215,264,94,175,219,10255,110,251,199,189,171,144,231,91,115,18362,112,73,257,101,232,247,71,259,114,6866,213,160,122,120,161,121,136,140,196,11193,233,103,206,137,128,203,227,117,205,204
C ≔ FrobeniusComplement⁡G
C≔1,652,1504,735,846,1327,268,1449,14110,15711,9812,10513,8914,14515,23816,21717,8218,23919,22620,26121,7622,22223,20624,12327,11829,26230,9731,3532,26433,12534,4436,13137,7738,10739,4340,15941,19742,23446,20747,18048,11249,10950,15351,14952,7153,21254,11355,10456,18357,15658,10359,19960,18561,23562,12663,13564,16866,13667,18668,21669,8870,11972,16474,24575,9978,22979,25580,26581,18183,24285,20986,26087,17090,9192,20493,22494,19495,17496,237100,167101,190102,192106,108110,116111,140114,198115,243117,154120,122121,213124,253127,151128,256129,147130,179133,188134,165137,252138,231139,178142,173143,227148,257152,249155,228158,214160,161162,266163,189166,219171,191175,246176,232177,223182,203184,218187,205193,210195,259200,230201,247202,263208,233215,248220,250221,236225,241244,251254,258,1,194,218,249,2322,97,100,220,1863,25,169,45,1724,44,49,15,465,157,40,185,666,111,212,105,1797,14,52,173,1758,104,243,116,2449,259,76,178,22210,159,60,136,8411,214,253,31,20212,130,132,140,5313,102,43,41,12616,156,114,85,16617,165,123,122,6118,48,153,215,13119,188,168,69,16120,174,260,74,8721,139,22,141,19523,208,252,256,18724,120,235,82,13426,145,71,142,24627,181,127,54,8328,196,211,240,14629,225,201,258,7830,167,250,67,15032,33,236,257,16434,109,238,207,7335,263,98,158,12436,239,112,50,24837,162,190,255,23438,228,223,135,6839,197,62,89,19242,77,266,101,7947,96,265,119,5155,115,110,251,14456,191,138,163,5957,198,209,219,21758,154,92,143,18263,216,107,155,17764,88,160,226,13365,94,184,152,17670,149,180,237,8072,264,125,221,14875,193,230,108,12181,151,113,242,11886,245,170,261,9599,210,200,106,213103,117,204,227,203128,205,206,233,137171,231,189,199,183229,262,241,247,254
IsMalnormal⁡C,G
The Frobenius complement in a Frobenius dihedral group is a subgroup of order two.
G ≔ DihedralGroup⁡7:
C≔1,62,53,4
The Mathieu group of degree 10 has a point stabilizer of order 72. (This is sometimes referred to as a Mathieu group of degree 9.)
G ≔ MathieuGroup⁡10
G≔M10
S ≔ Stabilizer⁡1,G
S≔2,8,4,63,9,5,10,2,6,7,93,5,8,10
GroupOrder⁡S
72
This point stabilizer is a Frobenius group.
IsFrobeniusGroup⁡S
Moreover, the action is Frobenius.
IsFrobeniusPermGroup⁡S
The Frobenius complement in S is a quaternion group.
AreIsomorphic⁡FrobeniusComplement⁡S,QuaternionGroup⁡
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.
See Also
GroupTheory[IsNilpotent]
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