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Action of a Frobenius-like group

Güloǧlu, Ismail Ş.
Ercan, Gülin
We call a finite group Frobenius-like if it has a nontrivial nilpotent normal subgroup F possessing a nontrivial complement H such that [F, h] = F for all nonidentity elements h is an element of H. We prove that any irreducible nontrivial FH-module for a Frobenius-like group FH of odd order over an algebraically closed field has an H-regular direct summand if either F is fixed point free on V or F acts nontrivially on V and the characteristic of the field is coprime to the order of F. Some consequences of this result are also derived.