- Frobenius group
In
mathematics , a Frobenius group is a transitivepermutation group on afinite set , such that no non-trivial elementfixes 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 anormal subgroup called the Frobenius kernel "K". (This is a theorem due to Frobenius.) The Frobenius group "G" is thesemidirect product of "K" and "H"::"G" = "K" ⋊ "H".Both the Frobenius kernel and the Frobenius complement have very restricted structures. J. G. Thompson proved that the Frobenius kernel "K" is a
nilpotent group . If "H" has even order then "K" is abelian. The Frobenius complement "H" has the property that every subgroup whose order is the product of 2 primes is cyclic; this implies that itsSylow subgroup s are cyclic or generalized quaternion groups. Any group such that all Sylow subgroups are cyclic is ametacyclic group : this means it is the extension of two cyclic groups. If a Frobenius complement "H" is not solvable then Zassenhaus showed that ithas 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 theSchur-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 invertibleaffine transformation s , acting naturally on "Fq" is a Frobenius group. The preceding example corresponds to the case "F3", the field with three elements.*Another example is provided by the subgroup of order 21 of the collineation group of the
Fano plane generated by a 3-fold symmetry σ fixing a point and a cyclic permutation τ of all 7 points, satisfying στ =τ²σ. Identifying "F"8* with the Fano plane, σ can be taken to be the restriction of theFrobenius automorphism σ("x")="x"² of "F"8 and τ to be multiplication by any element not in the prime field "F"2 (i.e. a generator of the cyclic multiplicative group of "F"8). This Frobenius group acts simply transitively on the 21 flags in the Fano plane, i.e. lines with marked points.*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 thesemidirect 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.
*The subgroup of a
Zassenhaus group fixing a point is a Frobenius group.*Frobenius groups whose Fitting subgroup has arbitrarily large nilpotency class were constructed by Ito: Let "q" be a prime power, "d" a positive integer, and "p" a prime divisor of "q" −1 with "d" ≤ "p". Fix some field "F" of order "q" and some element "z" of this field of order "p". The Frobenius complement "H" is the cyclic subgroup generated by the diagonal matrix whose "i,i"'th entry is "z""i". The Frobenius kernel "K" is the Sylow "q"-subgroup of GL("d","q") consisting of upper triangular matrices with ones on the diagonal. The kernel "K" has nilpotency class "d" −1, and the semidirect product "KH" is a 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 arestricted 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 correspondinginduced representation of "G" is also irreducible. These give the irreducible representations of "G" with "K" not in their kernel.Alternative definitions
There are a number of group theoretical properties which are interesting on their own right, but which happen to be equivalent to the group possessing a permutation representation that makes it a Frobenius group.
* "G" is a Frobenius group if and only if "G" has a proper, nonidentity subgroup "H" such that "H" ∩ "H""g" is the identity subgroup for every "g" ∈ "G" − "H".
This definition is then generalized to the study of trivial intersection sets which allowed the results on Frobenius groups used in the classification of
CA group s to be extended to the results onCN group s and finally the odd order theorem.Assuming that "G" = "K" ⋊ "H" is the
semidirect product of the normal subgroup "K" and complement "H", then the following restrictions oncentralizer s are equivalent to "G" being a Frobenius group with Frobenius complement "H":* The
centralizer C"G"("k") is a subgroup of K for every nonidentity "k" in "K"
* C"H"("k") = 1 for every nonidentity "k" in "K"
* C"G"("h") ≤ H for every nonidentity "h" in HReferences
*D. S. Passman, "Permutation groups", Benjamin 1968
*I. M. Isaacs, "Character theory of finite groups", AMS Chelsea 1976
*B. Huppert, "Endliche Gruppen I", Springer 1967
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