Frobenius groups of low rank

Let G be a finite Frobenius group of degree n. We show, by elementary means, that n is a power of some prime p provided the rank $${\mathrm{rk}}(G)\le 3+\sqrt{n+1}$$ rk ( G ) ≤ 3 + n + 1 . Then the Frobenius kernel of G agrees with the (unique) Sylow p-subgroup of G. So our result implies the celebrated theorems of Frobenius and Thompson in a special situation.

On Periodic Groups Saturated with Finite Frobenius Groups

A group is called weakly conjugate biprimitively finite if each its element of prime order generates a finite subgroup with any of its conjugate elements. A binary finite group is a periodic group in which any two elements generate a finite subgroup. If $\mathfrak{X}$ is some set of finite groups, then the group $G$ saturated with groups from the set $\mathfrak{X}$ if any finite subgroup of $G$ is contained in a subgroup of $G$, isomorphic to some group from $\mathfrak{X}$. A group $G = F \leftthreetimes H$ is a Frobenius group with kernel $F$ and a complement $H$ if $H \cap H^f = 1$ for all $f \in F^{\#}$ and each element from $G \setminus F$ belongs to a one conjugated to $H$ subgroup of $G$. In the paper we prove that a saturated with finite Frobenius groups periodic weakly conjugate biprimitive finite group with a nontrivial locally finite radical is a Frobenius group. A number of properties of such groups and their quotient groups by a locally finite radical are found. A similar result was obtained for binary finite groups with the indicated conditions. Examples of periodic non locally finite groups with the properties above are given, and a number of questions on combinatorial group theory are raised.

Infinite Frobenius Groups Generated by Elements of Order 3

A semidirect product [Formula: see text] of groups F and H is called a Frobenius group if the following two conditions are satisfied: (F1) H acts freely on F, that is, [Formula: see text] for f in F and h in H only if h = 1 or f = 1. (F2) Every non-identity element [Formula: see text] of finite order n induces in F by conjugation in G a splitting automorphism, that is, [Formula: see text] for every [Formula: see text]; in other words, the order of [Formula: see text] is equal to n. We describe the normal structure of a Frobenius group with periodic subgroup H generated by elements of order 3.

A note on modular Frobenius groups

It is well known that Frobenius groups can be defined by their complement subgroups. But until now we cannot use a complement subgroup to define a modular Frobenius group. In the present paper, a generalization of Frobenius complements is used as a characterization of a class of modular Frobenius groups. In fact, we build a connection between modular Frobenius groups and Frobenius–Wielandt groups.

Frobenius REA-groups and their Cayley graphs

In this paper, we determine the automorphism groups of a class of Frobenius groups, and then solve that under what condition they are REA-groups. As an application, we construct a type of normal edge-transitive Cayley graph.