Frobenius Group Research Articles

Sharp permutation groups whose point stabilizers are Frobenius groups with cyclic Frobenius kernel

Let ( G , X ) be a transitive non-geometric sharp permutation group of type { 0 , k } and let x ∈ X . We prove that if the point stabilizer G x is a Frobenius group with cyclic Frobenius kernel, then G x ≅ AGL ( 1 , p ) for some odd prime p and ( G , X ) is permutation isomorphic to a certain subgroup of AGL ( 2 , p ) acting on its natural module.

Frobenius Distributions of Low Dimensional Abelian Varieties Over Finite Fields

Abstract Given a $g$-dimensional abelian variety $A$ over a finite field $\mathbf{F}_{q}$, the Weil conjectures imply that the normalized Frobenius eigenvalues generate a multiplicative group of rank at most $g$. The Pontryagin dual of this group is a compact abelian Lie group that controls the distribution of high powers of the Frobenius endomorphism. This group, which we call the Serre–Frobenius group, encodes the possible multiplicative relations between the Frobenius eigenvalues. In this article, we classify all possible Serre–Frobenius groups that occur for $g \le 3$. We also give a partial classification for simple ordinary abelian varieties of prime dimension $g\geq 3$.

Class-preserving Coleman automorphisms of Frobenius groups

Class-preserving Coleman automorphisms of Frobenius groups

Group-subgroup subfactors revisited

For all Frobenius groups and a large class of finite multiply transitive permutation groups, we show that the corresponding group-subgroup subfactors are completely characterized by their principal graphs. The class includes all the sharply k -transitive permutation groups for k=2,3,4 , and in particular the Mathieu group M_{11} of degree 11.

О прямых произведениях групп диэдра в локально конечных группах

When studying infinite groups, as a rule, some finiteness conditions are imposed. For example, they require that the group be periodic, a Shunkov group, a Frobenius group, or a locally finite group. The concept of saturation allows us to effectively establish the internal structure of various classes of infinite groups. To date, a large array of results on groups saturated with various classes of finite groups has been obtained. Another important direction in the study of groups with saturation conditions is the study of groups saturated by direct products of various groups. Significant progress has been made in solving the problem of B. Amberg and L. S. Kazarin on periodic groups saturated with dihedral groups in the class of locally finite groups. It is proved that a locally finite group saturated by the direct product of a finite number of finite groups of a dihedron is isomorphic to the direct product of locally cyclic groups multiplied by an involution. It is also proved that a locally finite group saturated by the direct product of a finite number of finite dihedral groups is solvable.

Nilpotent residual of a finite group

Let F be a nilpotent group acted on by a group H via automorphisms and let the group G admit the semidirect product FH as a group of automorphisms so that CG(F)=1. We prove that the order of γ∞(G), the rank of γ∞(G) are bounded in terms of the orders of γ∞(CG(H)) and H, the rank of γ∞(CG(H)) and the order of H, respectively in cases where either FH is a Frobenius group; FH is a Frobenius-like group satisfying some certain conditions; or FH=〈α,β〉 is a dihedral group generated by the involutions α and β with F=〈αβ〉 and H=〈α〉.

Mock hyperbolic reflection spaces and Frobenius groups of finite Morley rank

Mock hyperbolic reflection spaces and Frobenius groups of finite Morley rank

The commuting conjugacy class graphs of finite groups with a given property

Abstract Let G be a finite non-abelian group. The commuting conjugacy class graph Γ ⁢ ( G ) {\Gamma(G)} is defined as a graph whose vertices are non-central conjugacy classes of G and two distinct vertices X and Y in Γ ⁢ ( G ) {\Gamma(G)} are connected by an edge if there exist elements x ∈ X {x\in X} and y ∈ Y {y\in Y} such that x ⁢ y = y ⁢ x {xy=yx} . In this paper, the structure of the commuting conjugacy class graph of group G with the property that G Z ⁢ ( G ) {\frac{G}{Z(G)}} is isomorphic to a Frobenius group of order pq or p 2 ⁢ q {p^{2}q} , is determined.

Finite solvable groups with a nilpotent normal complement subgroup

Let [Formula: see text] be a finite solvable group and [Formula: see text] a non-normal core-free solvable subgroup of [Formula: see text]. We show that if the normalizer of any nontrivial normal subgroup of [Formula: see text] is equal to [Formula: see text], then [Formula: see text] has a nilpotent normal complement [Formula: see text] such that [Formula: see text] and [Formula: see text] is a Frobenius group.

A note on Flavell’s theorem associated with Frobenius groups

Let G be a Frobenius group with the Frobenius kernel K. Suppose that G contains a nontrival subgroup D ⊆ K such that the normalizer N G ( D ) ⊆ K . When D is not a 2-group, Flavell proved, without using character theory, that K is a subgroup of G. Based on this result, we further prove that K is a subgroup when D is a 2-group.

Nilpotency of Lie Type Algebras with Metacyclic Frobenius Groups of Automorphisms

Nilpotency of Lie Type Algebras with Metacyclic Frobenius Groups of Automorphisms

Lie algebras with Frobenius dihedral groups of automorphisms

Suppose that a Lie algebra L admits a finite Frobenius group of automorphisms FH with cyclic kernel F and complement H of order 2 such that the fixed-point subalgebra of F is trivial and the fixed-point subalgebra of H is metabelian. Then L is solvable of derived length bounded by a constant.

Mutually Orthogonal Latin Squares as Group Transversals

Abstract In this paper, we give a method to determine a complete set of mutually orthogonal Latin squares of order m, where m is an odd prime or power of a prime, as a group transversal of a Frobenius group.

Extending Camina pairs

Let G be a finite group and N a nontrivial proper normal subgroup of G. A.R. Camina introduced the class of finite groups G, which extends Frobenius groups, satisfying that for all g∈G−N and n∈N, gn is conjugate to g. He proved that under these assumptions one of three possibilities occurs: G is a Frobenius group with kernel N; or N is a p-group; or G/N is a p-group. In this paper we extend this class of groups by investigating the structure of those finite groups G having a nontrivial proper normal subgroup N such that gn is conjugate to either g or g−1 for all g∈G−N and all n∈N.

Conjugacy classes of maximal cyclic subgroups

Abstract In this paper, we study the number of conjugacy classes of maximal cyclic subgroups of a finite group 𝐺, denoted η ⁢ ( G ) \eta(G) . First we consider the properties of this invariant in relation to direct and semi-direct products, and we characterize the normal subgroups 𝑁 with η ⁢ ( G / N ) = η ⁢ ( G ) \eta(G/N)=\eta(G) . In addition, by applying the classification of finite groups whose nontrivial elements have prime order, we determine the structure of G / ⟨ G − ⟩ G/\langle G^{-}\rangle , where G − G^{-} is the set of elements of 𝐺 generating non-maximal cyclic subgroups of 𝐺. More precisely, we show that G / ⟨ G − ⟩ G/\langle G^{-}\rangle is either trivial, elementary abelian, a Frobenius group or isomorphic to A 5 A_{5} .

Локально конечные группы, насыщенные прямым произведением двух конечных групп диэдра

In the study of infinite groups, as a rule, some finiteness conditions are imposed. For example, the group is required to be periodic, Shunkov group, Frobenius group, locally finite group. The concept of saturation makes it possible to effectively establish the internal structure of various classes of infinite groups. To date, a large array of results on groups saturated with various classes of finite groups has been obtained. Another important direction in the study of groups with saturation conditions is the study of groups saturated with direct products of various groups. In this paper, a partial solution to the problem of B. Amberg and L.S. Kazarin on periodic groups saturated with dihedral groups in the class of locally finite groups. The structure of a locally finite group saturated with a direct product of two finite dihedral groups is established and it is proved that in this case the group is solvable. The result obtained is an important step towards solving the problem of Amberg and Kazarin.

Classical groups as Frobenius complement

The Frobenius group G belongs to an important class of groups that more than 100 years ago was defined by F. G. Frobenius who proved that G is a semi-direct product of a normal subgroup K of G called kernel by another non-trivial subgroup H called the complement. In this case we show that a few of the classical finite groups can be Frobenius complement.

On pseudofrobenius imprimitive association schemes

An (association) scheme is said to be Frobenius if it is the (orbital) scheme of a Frobenius group. A scheme which has the same tensor of intersection numbers as some Frobenius scheme is said to be pseudofrobenius. We establish a necessary and sufficient condition for an imprimitive pseudofrobenius scheme to be Frobenius. We also prove strong necessary conditions for existence of an imprimitive pseudofrobenius scheme which is not Frobenius. As a byproduct, we obtain a sufficient condition for an imprimitive Frobenius group G with abelian kernel to be determined up to isomorphism only by the character table of G. Finally, we prove that the Weisfeiler-Leman dimension of a circulant graph with n vertices and Frobenius automorphism group is equal to 2 unless $$n\in \{p,p^2,p^3,pq,p^2q\}$$ , where p and q are distinct primes.

The GFR problem and a theorem of Godsil

A Frobenius group is a semi-direct product K ⋊ H with the property that the complement H acts fixed-point-freely on the non-trivial elements of the kernel K. A Cayley graph Γ = Cay(K,S) for the kernel K is called a graphical Frobenius representation (or GFR) of the Frobenius group G = K ⋊ H if G is the automorphism group of Γ. About five years ago, the second author of this paper conjectured (largely on the basis of a range of theoretical and computational observations made by the first author) that for any fixed finite group H, all but finitely many Frobenius groups with complement H have a GFR, except in cases where |H| is odd and the kernel K is abelian but is not an elementary 2-group. This conjecture was subsequently proved by Pablo Spiga, but it leaves an interesting problem which we call the GFR Problem, namely of determining the finitely many exceptions for a given group H. In this paper, we show what happens in a number of cases. For example, we prove that if K is isomorphic to ℤp 2 for an odd prime p, then no Frobenius group G = K ⋊ H has a GFR when H contains an element of order divisible by p + 1, while if K is isomorphic to ℤp n for some odd prime p and H acts on K as scalar multiplication, the only exceptions to having a GFR occur when n = 2 and p ∈ {3, 5, 7, 11, 13}, or (n,p) = (3,3).

Classification of minimal Frobenius hypermaps

In this paper, we give a classification of orientably regular hypermaps with an automorphism group that is a minimal Frobenius group. A Frobenius group G is called minimal if it has no nontrivial normal subgroup N such that G/N is a Frobenius group. An orientably regular hypermap H is called a Frobenius hypermap if Aut(H) acting on the hyperfaces is a Frobenius group. A minimal Frobenius hypermap is a Frobenius hypermap whose automorphism group is a minimal Frobenius group with cyclic point stabilizers. Every Frobenius hypermap covers a minimal Frobenius hypermap. The main theorem of this paper generalizes the main result of Breda D’Azevedo and Fernandes in 2011.