Frobenius Group Of Automorphisms Research Articles

Frobenius action on Carter subgroups

Let [Formula: see text] be a finite solvable group and [Formula: see text] be a subgroup of [Formula: see text]. Suppose that there exists an [Formula: see text]-invariant Carter subgroup [Formula: see text] of [Formula: see text] such that the semidirect product [Formula: see text] is a Frobenius group with kernel [Formula: see text] and complement [Formula: see text]. We prove that the terms of the Fitting series of [Formula: see text] are obtained as the intersection of [Formula: see text] with the corresponding terms of the Fitting series of [Formula: see text], and the Fitting height of [Formula: see text] may exceed the Fitting height of [Formula: see text] by at most one. As a corollary it is shown that for any set of primes [Formula: see text], the terms of the [Formula: see text]-series of [Formula: see text] are obtained as the intersection of [Formula: see text] with the corresponding terms of the [Formula: see text]-series of [Formula: see text], and the [Formula: see text]-length of [Formula: see text] may exceed the [Formula: see text]-length of [Formula: see text] by at most one. These theorems generalize the main results in [E. I. Khukhro, Fitting height of a finite group with a Frobenius group of automorphisms, J. Algebra 366 (2012) 1–11] obtained by Khukhro.

Frobenius groups of automorphisms with almost fixed point free kernel

Let FH be a Frobenius group with kernel F and complement H, acting coprimely on the finite solvable group G by automorphisms. We prove that if CG(H) is of Fitting length n then the index of the n-th Fitting subgroup Fn(G) in G is bounded in terms of |CG(F)| and |F|. This generalizes a result of Khukhro and Makarenko [6] which handles the case n=1.

Finite 𝑝-groups with a Frobenius group of automorphisms whose kernel is a cyclic 𝑝-group

Suppose that a finite p p -group P P admits a Frobenius group of automorphisms F H FH with kernel F F that is a cyclic p p -group and with complement H H . It is proved that if the fixed-point subgroup C P ( H ) C_P(H) of the complement is nilpotent of class c c , then P P has a characteristic subgroup of index bounded in terms of c c , | C P ( F ) | |C_P(F)| , and | F | |F| whose nilpotency class is bounded in terms of c c and | H | |H| only. Examples show that the condition of F F being cyclic is essential. The proof is based on a Lie ring method and a theorem of the authors and P. Shumyatsky about Lie rings with a metacyclic Frobenius group of automorphisms F H FH . It is also proved that P P has a characteristic subgroup of ( | C P ( F ) | , | F | ) (|C_P(F)|, |F|) -bounded index whose order and rank are bounded in terms of | H | |H| and the order and rank of C P ( H ) C_P(H) , respectively, and whose exponent is bounded in terms of the exponent of C P ( H ) C_P(H) .

Nilpotent Length of a Finite Solvable Group with a Frobenius Group of Automorphisms

We prove that a finite solvable group G admitting a Frobenius group FH of automorphisms of coprime order with kernel F and complement H such that [G, F] = G and C C G (F)(h) = 1 for all nonidentity elements h ∈ H, is of nilpotent length equal to the nilpotent length of the subgroup of fixed points of H.

The dihedral group as a group of automorphisms

Suppose that D=〈α,β〉 is a dihedral group generated by two involutions α and β. Let D act on a finite group G in such a manner that CG(αβ)=1. We show that if CG(α) and CG(β) are both nilpotent of class c, then G is nilpotent and the class of G is bounded solely in terms of c. If both CG(α) and CG(β) are of exponent dividing e, then the exponent of G is bounded solely in terms of e and |D|. Previously, results of this kind were known only for groups acted on by Frobenius groups of automorphisms.

Exponent of a finite group with a four-group of automorphisms

Let A be a group isomorphic with either S4, the symmetric group on four symbols, or D8, the dihedral group of order 8. Let V be a normal four-subgroup of A and α an involution in \({A\setminus V}\). Suppose that A acts on a finite group G in such a manner that CG(V) = 1 and CG(α) has exponent e. We show that if \({A\cong S_4}\) then the exponent of G is e-bounded and if \({A\cong D_8}\) then the exponent of the derived group G′ is e-bounded. This work was motivated by recent results on the exponent of a finite group admitting an action by a Frobenius group of automorphisms.

Fixed points of the complements of Frobenius groups of automorphisms

Suppose that a finite group G admits a Frobenius group of automorphisms BA with kernel B and complement A. It is proved that if N is a BA-invariant normal subgroup of G such that (|N|, |B|) = 1 and C N (B) = 1 then C G/N (A) = C G (A)N/N. If N = G is a nilpotent group then we give as a corollary some description of the fixed points C L(G)(A) in the associated Lie ring L(G) in terms of C G (A). In particular, this applies to the case where GB is a Frobenius group as well (so that GBA is a 2-Frobenius group, with not necessarily coprime orders of G and A).

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 CG(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 CG(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 CG(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.

On symmetric (69,17,4)-designs admitting an action of Frobenius groups

Abstract The aim of this paper is to achieve the complete classification of (69,17,4)-designs admitting an action of any Frobenius group of automorphisms the kernel of which is isomorphic to one of the cyclic groups of order 13, 17 and 23. A positive result can be got only in the case of the nonabelian group of order 39, where the cyclic group of order 3 fixes exactly 9 points.