Narrow Results

Let $g$ be an element of a group $G$. For a positive integer $n$, let $E_n(g)$ be the subgroup generated by all commutators $[\dots [[x,g],g],\dots ,g]$ over $x\in G$, where $g$ is repeated $n$ times. We prove that if $G$ is a profinite group such that for every $g\in G$ there is $n=n(g)$ such that $E_n(g)$ is finite, then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent. The proof uses the Wilson–Zelmanov theorem saying that Engel profinite groups are locally nilpotent. In the case of a finite group $G$, we prove that if, for some $n$, $|E_n(g)|\le m$ for all $g\in G$, then the order of the nilpotent residual ${\gamma }_{\infty }(G)$ is bounded in terms of $m$.

Let ℨ be a complete set of Sylow subgroups of a finite group G, that is, for each prime p dividing the order of G, ℨ contains exactly one and only one Sylow p-subgroup of G, say G_p. Let C be a nonempty subset of G. A subgroup H of G is said to be C-ℨ-permutable (conjugate-ℨ-permutable) subgroup of G if there exists some x ∈ C such that H^xG_p = G_pH^x, for all G_p ∈ ℨ. We investigate the structure of the finite group G under the assumption that certain subgroups of prime power orders of G are C-ℨ-permutable (conjugate-ℨ-permutable) subgroups of G. Our results improve and generalize several results in the literature.

Let ℨ be a complete set of Sylow subgroups of a finite group G, that is, for each prime p dividing the order of G, ℨ contains exactly one and only one Sylow p-subgroup of G. A subgroup H of G is said to be ℨ-permutable of G if H permutes with every member of ℨ. A subgroup H of G is said to be a weakly ℨ-permutable subgroup of G if there exists a subnormal subgroup K of G such that G = HK and H ∩ K ≤ H_ℨ, where H_ℨ is the subgroup of H generated by all those subgroups of H which are ℨ-permutable subgroups of G. In this paper, we prove that if p is the smallest prime dividing the order of G and the maximal subgroups of G_p ∈ ℨ are weakly ℨ-permutable subgroups of G, then G is p-nilpotent. Moreover, we prove that if 𝔉 is a saturated formation containing the class of all supersolvable groups, then G ∈ 𝔉 iff there is a solvable normal subgroup H in G such that G/H ∈ 𝔉 and the maximal subgroups of the Sylow subgroups of the Fitting subgroup F(H) are weakly ℨ-permutable subgroups of G. These two results generalize and unify several results in the literature.

Let $\mathrm{\text{cd}}(G)$ be the set of all irreducible complex characters of a finite group $G$. In [K. Aziziheris, Determining group structure from sets of irreducible character degrees, J. Algebra323 (2010) 1765–1782], we proved that if $m$ and $n$ are relatively prime integers greater than $1$, $p$ is prime not dividing $mn$, and $G$ is a solvable group such that $\mathrm{\text{cd}}(G)=\{1,p,n,m,pn,pm\}$, then under some conditions on $p,m,$ and $n$, the group $G=A\times B$ is the direct product of two normal subgroups, where $\mathrm{\text{cd}}(A)=\{1,p\}$ and $\mathrm{\text{cd}}(B)=\{1,n,m\}$. In this paper, we replace $p$ by $u$, where $u>1$ is an arbitrary positive integer, and we obtain similar result. As an application, we show that if $G$ is a finite group with $\mathrm{\text{cd}}(G)=\{1,21,13,55,273,1155\}$ or $\mathrm{\text{cd}}(G)=\{1,15,391,58,5865,870\}$, then $G$ is a direct product of two non-abelian normal subgroups.

Finite groups with an automorphism mapping a sufficiently large proportion of elements to their inverses, squares and cubes have been studied for a long time, and the gist of the results on them is that they are “close to being abelian”. In this paper, we consider finite groups $G$ such that, for a fixed but arbitrary $\rho \in (0,1]$, some automorphism of $G$ maps at least $\rho \left|G\right|$ many elements of $G$ to their inverses, squares and cubes. We will relate these conditions to some parameters that measure, intuitively speaking, how far the group $G$ is from being solvable, nilpotent or abelian; most prominently the commuting probability of $G$, i.e. the probability that two independently uniformly randomly chosen elements of $G$ commute. To this end, we will use various counting arguments, the classification of the finite simple groups and some of its consequences, as well as a classical result from character theory.

Let m and n be positive integer numbers. In this paper, we study $𝔑(m,n)$, the class of all groups G that for all subsets m and n of G containing m and n elements, respectively, there exist $x\in M$ and $y\in N$ such that $〈x,y〉$ is nilpotent, which introduced by Zarrin in 2012. We improve some results of Zarrin and find some sharp bounds for m and n such that $G\in 𝔑(m,n)$ implies that G is nilpotent. Also we will characterize all finite $𝔫$-groups in $𝔑(m,n)$, which $m+n\le 10$.

In this paper, we study the structure of finite groups G under the assumption that certain abelian subgroups of largest possible exponent of prime power order lie in the 𝔉-hypercenter of G. We improve and extend some results of Ezzat, Shaalan, Elashiry, Asaad, Ramadan, and Buckley.

A finite permutation group $G\le \mathrm{\text{Sym}}(\mathrm{\Omega })$ is called $2$-closed if $G$ is the largest subgroup of $\mathrm{\text{Sym}}(\mathrm{\Omega })$ which leaves invariant each of the $G$-orbits for the induced action on $\mathrm{\Omega }\times \mathrm{\Omega }$. Introduced by Wielandt in 1969, the concept of $2$-closure has developed as one of the most useful approaches for studying relations on a finite set invariant under a group of permutations of the set; in particular for studying automorphism groups of graphs and digraphs. The concept of total $2$-closure switches attention from a particular group action, and is a property intrinsic to the group: a finite group $G$ is said to be totally $2$-closed if $G$ is $2$-closed in each of its faithful permutation representations. There are infinitely many finite soluble totally $2$-closed groups, and these have been completely characterized, but up to now no insoluble examples were known. It turns out, somewhat surprisingly to us, that there are exactly $6$ totally $2$-closed finite nonabelian simple groups: the Janko groups ${\mathrm{\text{J}}}_1$, ${\mathrm{\text{J}}}_3$ and ${\mathrm{\text{J}}}_4$, together with $\mathrm{\text{Ly}}$, $\mathrm{\text{Th}}$ and the Monster $𝕄$. Moreover, if a finite totally $2$-closed group has no nontrivial abelian normal subgroup, then we show that it is a direct product of some (but not all) of these simple groups, and there are precisely $47$ examples.

In the course of obtaining this classification, we develop a general framework for studying $2$-closures of transitive permutation groups, which we hope will prove useful for investigating representations of finite groups as automorphism groups of graphs and digraphs, and in particular for attacking the long-standing polycirculant conjecture. In this direction, we apply our results, proving a dual to a 1939 theorem of Frucht from Algebraic Graph Theory. We also pose several open questions concerning closures of permutation groups.

In this paper, we study solvable groups in which $r_n(F)$ is at most 2. In particular, we investigated groups of odd order and $A_4$-free groups with this property. Exact estimations of the derived length and nilpotent length of such groups are obtained.