Generalized Frobenius groups

A pair (G. K) in whichG is a finite group andK◃G, 1<K<G, is said to satisfy (F2) if |C _G (x)|=|C _G/K (xK)| for allx∈G/K. First we survey all the examples known to us of such pairs in whichG is neither ap-group nor a Frobenius group with Frobenius kernelK. Then we show that under certain restrictions there are, essentially, all the possible examples.     

Double Frobenius groups

Double Frobenius groups are studied. Some properties of a minimal counterexample to V.D. Mazurov’s conjecture about these groups are obtained. Under some additional restrictions the conjecture is confirmed.     

Some infinite dimensional representations of reductive groups with Frobenius maps

In this paper,we construct certain irreducible infinite dimensional representations of algebraic groups with Frobenius maps.In particular,a few classical results of Steinberg and Deligne&Lusztig on complex representations of finite groups of Lie type are extended to reductive algebraic groups with Frobenius maps.     

Sufficient conditions for the almost layer finiteness of groups

We prove a theorem that describes almost layer-finite groups in the class of conjugatively biprimitive-finite groups. Computer Center of the Siberian Division of the Russian Academy of Sciences, Krasnoyarsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 4, pp. 472–485, April, 1999.     

Frobenius groups and retract rationality

Let k$k$ be any field, G$G$ be a finite group acting on the rational function field k(_xg:g∈G)$k(x_g:g\in G)$ by h⋅_xg=_xhg$h\cdot x_g=x_{hg}$ for any h,g∈G$h,g\in G$. Define k(G)=k(_xg:g∈G)^G$k\left(G\right)=k{(x_g:g\in G)}^G$. Noether’s problem asks whether k(G)$k\left(G\right)$ is rational (= purely transcendental) over k$k$. A weaker notion, retract rationality introduced by Saltman, is also very useful for the study of Noether’s problem. We prove that, if G$G$ is a Frobenius group with abelian Frobenius kernel, then k(G)$k\left(G\right)$ is retract k$k$-rational for any field k$k$ satisfying some mild conditions. As an application, we show that, for any algebraic number field k$k$, for any Frobenius group G$G$ with Frobenius complement isomorphic to S_L2(_F5)$S{L}_2\left({\mathbb{F}}_5\right)$, there is a Galois extension field K$K$ over k$k$ whose Galois group is isomorphic to G$G$, i.e. the inverse Galois problem is valid for the pair (G,k)$(G,k)$. The same result is true for any non-solvable Frobenius group if k(_ζ8)$k\left({\zeta }_8\right)$ is a cyclic extension of k$k$.     

Finite groups with Frobenius subgroup

Suppose the normalizer N of a subgroup A of a simple group G is a Frobenius group with kernel A, and the intersection of A with any other conjugate subgroup of G is trivial, and suppose, if A is elementary Abelian, that ¦a¦> 2n+1, where n=¦N:A¦. It is proved that if A has a complement B in G, then G acts doubly transitively on the set of right cosets of G modulo B, the subgroup B is maximal in G, and ¦B¦ is divisible by ¦a¦–1. The proof makes essential use of the coherence of a certain set of irreducible characters of N.Translated from Matematicheskie Zametki, Vol. 20, No. 2, pp. 177–186, August, 1976.The author would like to thank V. D. Mazurov for helpful discussions concerning the theorem proved in this note.     

Monomiality conditions for Frobenius group

A monomiality criterion for a Frobenius group in terms of its complement is proved. In addition, a sufficient monomality conditions for a Frobenius group based on elementary properties of its kernel and complement is obtained.     

Frobenius Galois groups over quadratic fields

There exists a quadratic fieldQ(√D) over which every Frobenius group is realizable as a Galois group.     

Semi-regular divisible designs and Frobenius groups

We study and characterize semi-regular (s, k, λ₁, λ₂)-divisible designs which admit a Frobenius group as their translation group. Moreover, we give a construction method for such designs by generalized admissible triads.     

Nilpotency of finite groups with Frobenius groups of automorphisms

Suppose that a finite group G admits a Frobenius group of automorphisms BC of coprime order with kernel B and complement C such that C _G (C) is abelian. It is proved that if B is abelian of rank at least two and [C_G(u), C_G(v),...,C_G(v)]=1{[C_G(u), C_G(v),dots,C_G(v)]=1} for any u,v ? B{1}{u,vin B{setminus}{1}}, where C _G (v) is repeated k times, then G is nilpotent of class bounded in terms of k and |C| only. It is also proved that if B is abelian of rank at least three and C _G (b) is nilpotent of class at most c for every b ? B{1}{b in B{setminus}{1}}, then G is nilpotent of class bounded in terms of c and |C|. The proofs are based on results on graded Lie rings with many commuting components.     

Nilpotency of finite groups with Frobenius groups of automorphisms

Suppose that a finite group G admits a Frobenius group of automorphisms BC of coprime order with kernel B and complement C such that C _G (C) is abelian. It is proved that if B is abelian of rank at least two and \({[C_G(u), C_G(v),\dots,C_G(v)]=1}\) for any \({u,v\in B{\setminus}\{1\}}\), where C _G (v) is repeated k times, then G is nilpotent of class bounded in terms of k and |C| only. It is also proved that if B is abelian of rank at least three and C _G (b) is nilpotent of class at most c for every \({b \in B{\setminus}\{1\}}\), then G is nilpotent of class bounded in terms of c and |C|. The proofs are based on results on graded Lie rings with many commuting components.     

Supersolvable Frobenius groups with nilpotent centralizers

Let FH be a supersolvable Frobenius group with kernel F and complement H. Suppose that a finite group G admits FH as a group of automorphisms in such a manner that $C_G(F)=1$ and $C_G(H)$ is nilpotent of class c. We show that G is nilpotent of $(c,\left|FH\right|)$-bounded class.