Frobenius group
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Structure
Suppose G is a Frobenius group consisting of permutations of a set X. A subgroup H of G fixing a point of X is called a 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 (1901); there is still no proof of this theorem that does not use character theory, although see [1].) The Frobenius group G is the semidirect product of K and H:
- .
Both the Frobenius kernel and the Frobenius complement have very restricted structures.
The Frobenius kernel K is uniquely determined by G as it is the
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 transformations , 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 simply transitively on the 21 flagsin the Fano plane, i.e. lines with marked points.
- The dihedral group of order 2n 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 K1.H and K2.H then (K1 × K2).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. This was the first example of Frobenius group with nonabelian kernel (it was constructed by Otto Schmidt).
- If H is the group SL2(F5) 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 zi. 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 representations of G:
- Any irreducible representation R of H gives an irreducible representation of G using the quotient map from G to H. 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.
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 ∩ Hg is the identity subgroup for every g ∈ G − H, i.e. H is a malnormal subgroup of G.
This definition is then generalized to the study of trivial intersection sets which allowed the results on Frobenius groups used in the classification of
Assuming that is the
- The centralizerCG(k) is a subgroup of K for every nonidentity k in K.
- CH(k) = 1 for every nonidentity k in K.
- CG(h) ≤ H for every nonidentity h in H.
References
- Frobenius, G. (1901), "Über auflösbare Gruppen. IV.", Berl. Ber. (in German): 1216–1230, JFM 32.0137.01
- B. Huppert, Endliche Gruppen I, Springer 1967
- I. M. Isaacs, Character theory of finite groups, AMS Chelsea 1976
- D. S. Passman, Permutation groups, Benjamin 1968
- MR 0117289