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Let 𝐴 be a finite group acting by automorphisms on the finite group 𝐺. We introduce the commuting graph Γ(G, A) of this action and study some questions related to the structure of 𝐺 under certain graph theoretical conditions on Γ(G, A). [ABSTRACT FROM AUTHOR]

In this paper, we classify all finite groups satisfying the following property: their conjugacy class lengths are set-wise relatively prime for any six distinct classes. [ABSTRACT FROM AUTHOR]

We call a finite group Frobenius-like if it has a nontrivial nilpotent normal subgroup F possessing a nontrivial complement H such that for all nonidentity elements h ∈ H. We prove that any irreducible nontrivial F H-module for a Frobenius-like group F H 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. [ABSTRACT FROM AUTHOR]

Erratum to . [ABSTRACT FROM AUTHOR]