Narrow Results

Let $G$ be a finite solvable group and $H$ be a subgroup of $\mathrm{\text{Aut}}(G)$. Suppose that there exists an $H$-invariant Carter subgroup $F$ of $G$ such that the semidirect product $FH$ is a Frobenius group with kernel $F$ and complement $H$. We prove that the terms of the Fitting series of $C_G(H)$ are obtained as the intersection of $C_G(H)$ with the corresponding terms of the Fitting series of $G$, and the Fitting height of $G$ may exceed the Fitting height of $C_G(H)$ by at most one. As a corollary it is shown that for any set of primes $\pi$, the terms of the $\pi$-series of $C_G(H)$ are obtained as the intersection of $C_G(H)$ with the corresponding terms of the $\pi$-series of $G$, and the $\pi$-length of $G$ may exceed the $\pi$-length of $C_G(H)$ by at most one. These theorems generalize the main results in [E. I. Khukhro, Fitting height of a finite group with a Frobenius group of automorphisms, J. Algebra366 (2012) 1–11] obtained by Khukhro.

Let $G$ and $A$ be groups where $A$ acts on $G$ by automorphisms. We say “the action of $A$ on $G$ is good” if the equality $H=[H,B]C_H(B)$ holds for any subgroup $B$ of $A$ and for any $B$-invariant subgroup $H$ of $G$. It is straightforward that every coprime action is a good action. In this work, we extend some results due to Ward, Gross, Shumyatsky, Jabara and Meng and Guo under coprime action to good action.

In this paper, we classify and determine the structure of finite groups whose character degree graphs are empty graphs.

In this article, we prove that a finite solvable group with character degree graph containing at least four vertices has Fitting height at most 4 if each derived subgraph of four vertices has total degree not more than 8. We also prove that if the vertex set ρ(G) of the character degree graph Δ(G) of a solvable group G is a disjoint union ρ(G) = π₁ ∪ π₂, where |π_i| ≥ 2 and p_i, q_i∈ π_i for i = 1,2, and no vertex in π₁ is adjacent in Δ(G) to any vertex in π₂ except for p₁p₂ and q₁q₂, then the Fitting height of G is at most 4.

We discuss applications of Clifford's theorem in the situation where a finite group G admits a Frobenius group of automorphisms F H with complement H and fixed-point-free kernel F (that is, such that C_G(F) = 1). The strong hypothesis makes it easier to apply Clifford's theorem, but the desired results must also have strong conclusions. The aim is proving that various properties of G are close to the corresponding properties of C_G(H). It is shown that the fixed points of H in any F H-invariant section are covered by the fixed points of H in G. This implies a bound for |G| in terms of |H| and |C_G(H)|, and a bound for the rank of G in terms of |H| and the rank of C_G(H). It is shown that G is nilpotent if C_G(H) is nilpotent. These results do not need any coprimeness assumptions on the orders of G and FH. Under the additional condition that (|G|, |H|) = 1 it is proved that the Fitting height of G is equal to the Fitting height of the fixed-point subgroup C_G(H) and the Fitting series of C_G(H) coincides with its intersections with the Fitting series of G. Further recent results on bounding the exponent and nilpotency class of G are mentioned, which use different Lie ring methods.