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fixes more than one point and some non-trivial element fixes a point.
They are named after F. G. Frobenius .


STRUCTURE

The Subgroup ''H'' of a Frobenius group ''G'' fixing a point of the set ''X'' is called the Frobenius complement. The identity element together with all elements not in any conjugate of ''H'' form a Normal Subgroup called the '''Frobenius kernel''' ''K''. (This is a theorem due to Frobenius.) The Frobenius group ''G'' is the Semidirect Product of ''K'' and ''H'':
G


Both the Frobenius kernel and the Frobenius complement have very restricted structures. showed that it
has a normal subgroup of index 1 or 2 that is the product of SL2(5) and a metacyclic group of order coprime to 30. If a Frobenius complement ''H'' is solvable then it has a normal metacyclic subgroup such that the quotient is a subgroup of the symmetric group on 4 points.

The Frobenius kernel ''K'' is uniquely determined by ''G'' as it is the Fitting Subgroup , and the Frobenius complement is uniquely determined up to conjugacy by the Schur-Zassenhaus Theorem . In particular a finite group ''G'' is a Frobenius group in at most one way.


EXAMPLES

  • The smallest example is the symmetric group on 3 points, with 6 elements. The Frobenius kernel ''K'' has order 3, and the complement ''H'' has order 2.


  • For every Finite Field ''Fq'' with ''q'' (> 2) elements, the group of invertible Affine Transformation s x \mapsto ax+b , a

    • e 0 with its natural action on ''Fq'' is a Frobenius group. The preceding example corresponds to the case ''F3'', the field with three elements.


  • More generally, the group of upper 2 × 2 invertible Triangular Matrices of determinant 1 over any finite field of order at least 3 is a Frobenius group. The Frobenius kernel is the subgroup of strictly upper triangular matrices (with diagonal elements equal to 1), and the complement is the subgroup of diagonal matrices.


  • The Dihedral Group of order 2''n'' with ''n'' odd is a Frobenius group with complement of order 2. More generally if ''K'' is any abelian group of odd order and ''H'' has order 2 and acts on ''K'' by inversion, then the Semidirect Product ''K.H'' is a Frobenius group.


  • Many further examples can be generated by the following constructions. If we replace the Frobenius complement of a Frobenius group by a non-trivial subgroup we get another Frobenius group. If we have two Frobenius groups ''K''1.''H'' and ''K''2.''H'' then (''K''1 × ''K''2).''H'' is also a Frobenius group.


  • If ''K'' is the non-abelian group of order 73 with exponent 7, and ''H'' is the cyclic group of order 3, then there is a Frobenius group ''G'' that is an extension ''K.H'' of ''H'' by ''K''. This gives an example of a Frobenius group with non-abelian kernel.


  • If ''H'' is the group ''SL''2(''F''5) of order 120, it acts fixed point freely on a 2-dimensional vector space ''K'' over the field with 11 elements. The extension ''K.H'' is the smallest example of a non- Solvable Frobenius group.




REPRESENTATION THEORY

The irreducible complex representations of a Frobenius group ''G'' can be read off from those of ''H'' and ''K''. There are two types of Irreducible Representation s of ''G'':
  • Any irreducible representation ''R'' of ''H'' gives an irreducible representation of ''G'' using the quotient map from ''G'' to ''H'' (that is, as a Restricted Representation ). These give the irreducible representations of ''G'' with ''K'' in their kernel.

  • If ''S'' is any ''non-trivial'' irreducible representation of ''K'', then the corresponding Induced Representation of ''G'' is also irreducible. These give the irreducible representations of ''G'' with ''K'' not in their kernel.



REFERENCES


  • D. S. Passman, ''Permutation groups'', Benjamin 1968