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Let A be a group acting by automorphisms on the group G. The commuting graph Γ (G , A) of A-orbits of this action is the simple graph with vertex set { x A : 1 ≠ x ∈ G } , the set of all A-orbits on G ∖ { 1 } , where two distinct vertices xA and yA are joined by an edge if and only if there exist x 1 ∈ x A and y 1 ∈ y A such that [ x 1 , y 1 ] = 1 . The present paper characterizes the groups G for which Γ (G , A) is an F -graph, that is, a connected graph which contains at most one vertex whose degree is not less than three. [ABSTRACT FROM AUTHOR]

Suppose that A is a finite nilpotent group of odd order having a good action, in the sense of [1], on the group G of odd order. Under some additional assumptions we prove that the Fitting height of G is bounded above by the sum of the numbers of primes dividing | A | and | C G (A) | counted with multiplicities. [ABSTRACT FROM AUTHOR]

A subgroup H of a group G is said to be a TNI-subgroup if for any Let A be an abelian group acting coprimely on the finite group G by automorphisms in such a way that for all is a solvable TNI-subgroup of G. We prove that G is a solvable group with Fitting length h(G) is at most. In particular whenever is nonnormal. Here, h(G) is the Fitting length of G and is the number of primes dividing A counted with multiplicities. [ABSTRACT FROM AUTHOR]

A subgroup H of a finite group G is called a TNI -subgroup if N G ( H ) ∩ H g = 1 for any g ∈ G \ N G ( H ) . Let A be a group acting on G by automorphisms where C G ( A ) is a TNI -subgroup of G . We prove that G is solvable if and only if C G ( A ) is solvable, and determine some bounds for the nilpotent length of G in terms of the nilpotent length of C G ( A ) under some additional assumptions. We also study the action of a Frobenius group FH of automorphisms on a group G if the set of fixed points C G ( F ) of the kernel F forms a TNI -subgroup, and obtain a bound for the nilpotent length of G in terms of the nilpotent lengths of C G ( F ) and C G ( H ) . [ABSTRACT FROM AUTHOR]

A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F with a nontrivial complement H such that $$ [F,h]=F$$ for all nonidentity elements $$h\in H$$ . Let FH be a Frobenius-like group with complement H of prime order such that $$C_F(H)$$ is of prime order. Suppose that FH acts on a finite group G by automorphisms where $$ (|G|,|H|)=1$$ in such a way that $$C_G(F)=1.$$ In the present paper we prove that the Fitting series of $$C_G(H)$$ coincides with the intersections of $$ C_G(H)$$ with the Fitting series of G, and the nilpotent length of G exceeds the nilpotent length of $$C_G(H)$$ by at most one. As a corollary, we also prove that for any set of primes $$\pi $$ , the upper $$\pi $$ -series of $$ C_G(H)$$ coincides with the intersections of $$C_G(H)$$ with the upper $$\pi $$ -series of G, and the $$\pi $$ - length of G exceeds the $$\pi $$ -length of $$ C_G(H)$$ by at most one. [ABSTRACT FROM AUTHOR]

We call the action of an automorphism α of a finite groupGa Hughes type action if it is described by conditions on the orders of elements ofG ⟨ α ⟩ − G. In the present paper we study the structure of finite groupGadmitting an automorphism α of prime orderpso that the orders of elements inG ⟨ α ⟩ − Gare not divisible byp2. [ABSTRACT FROM AUTHOR]

We prove that a finite solvable groupGadmitting a Frobenius groupFHof automorphisms of coprime order with kernelFand complementHsuch that [G,F] = GandCC G (F)(h) = 1 for all nonidentity elementsh ∈ H, is of nilpotent length equal to the nilpotent length of the subgroup of fixed points ofH. [ABSTRACT FROM AUTHOR]