Action of a Frobenius-like group

Action of a Frobenius-like group with fixed-point free kernel

“…We see that these results are essentially due to the fact that any FH -module V on which F acts fixed-point freely is a free H -module. In [3] we have been able to observe that not exactly the freeness, but a result close to this, is also true for Frobenius-like groups of odd order. For the sake of easy reference we restate below the main result in [3].…”

“…It is well known that Frobenius actions are coprime actions and the kernel F is nilpotent. In [3] a semidirect product with normal nilpotent subgroup F and complement H was called a Frobenius-like group with kernel F and complement H if the condition C F .h/ D 1 is replaced by OEF; h D F for all nonidentity elements h 2 H . Observe that H has the structure of a Frobenius complement and the orders of F and H are coprime since .FH /=F 0 is a Frobenius group and .F / D .F=F 0 /.…”

Frobenius-like groups as groups of automorphisms

“…It is a natural and important problem to extend these results to more general situations, both from the viewpoint of relaxing the strong conditions on the action of the kernel and relaxing the conditions on the structure of the group F H itself. Focusing on the Fitting height and related parameters, Ercan and Güloglu introduced the concept of a Frobenius-like group and obtained the results presented in [2,3], and together with Khukhro the results in [4].…”

“…However, the work by Ercan and Güloglu in [2, Theorem A] shows that it is not very far from being free, at least for certain Frobenius-like groups, in the sense that it contains a regular kH -module that guarantees that C V (H) is nontrivial. Namely, Theorem A in [2] is proved by reducing the structure of a minimal counterexample to a very restricted configuration and deducing a contradiction by proving the following theorem [2, Proposition C] on representations of some specific groups having a normal extraspecial subgroup, which is also of independent interest. We now consider the following complicated-looking hypothesis introduced in [4].…”

“…We now consider the following complicated-looking hypothesis introduced in [4]. It is formulated to avoid the so-called exceptional cases, which possibly occur in Hall-Higman type arguments, and is slightly more general than assuming that F H is of odd order as in the hypothesis of Theorem A in [2]. …”