Narrow Results

Let $G$ be a finite group. A subgroup $H$ of $G$ is called Hall normally embedded in $G$ if $H$ is a Hall subgroup of $H^G$, where $H^G$ is the normal closure of $H$ in $G$, that is, the smallest normal subgroup of $G$ containing $H$. A group $G$ is called a $PHN$-group if its all minimal subgroups and cyclic subgroup of order $4$ are Hall normally embedded in $G$. In this paper, we give the classification of minimal non-$PHN$-groups. Furthermore, we investigate the structure of finite group all of whose maximal subgroups of even order are $PHN$-groups.

A proper subgroup $H$ of a finite group $G$ is called a second maximal subgroup of $G$ if $H$ is a maximal subgroup of every maximal subgroup $M$ in $G$ with $H\le M$. In this paper, we investigate the structure of the finite group $G$ in which $C_G(x)$ is a second maximal subgroup for every non-central element $x$ in $G$, and prove that either $G$ is solvable or $G/Z(G)\cong \mathrm{\text{PSL}}(2,r^n)$, where $r=2$ with $n$ a prime or $r=3$ with $n$ an odd prime.

We investigate the interplay between the algebraic structure of a group G and arithmetic properties of its spectrum σ(G) which consists of the eigenvalues of all the inner automorphisms of G. A complex number λ is called an eigenvalue of a group automorphism A:G→G if φ◦A|H=λ·φ for some non-trivial homomorphism defined on an A-invariant subgroup H⊂G.

It is shown that many properties of a group G (such as the presence of a finitely generated subgroup of infinite rank, nilpotence, periodicity, polycyclicity, etc.) are coded in its spectrum. In this paper the spectra are applied to the investigation of the so-called reversive properties of groups. The paper ends with a list of related open problems.

Let G be a group and G′ be its commutator subgroup. Denote by c(G) the minimal number such that every element of G′ can be expressed as a product of at most c(G) commutators. We find suitable bounds for c(G) when G is a free nilpotent by abelian group. Then we prove that c(G) is finite if G is a n-generator solvable group. And G has a nilpotent by abelian normal subgroup K of finite index. Moreover we have c(G) ≤ s(s + 1)/2 + 72n² + 47n, where s is the number of generators of K. We also prove that in a solvable group of finite Pruffer rank s every element of its commutator subgroup is equal to a product of at most s(s + 1)/2 + 72s² + 47s. And finally as a corollary of the above results we show that if A is a normal subgroup of a solvable group G such that G/A is a d-generator finite group. And A has finite Pruffer rank s. Then c(G) ≤ s(s + 1)/2 + 72(s² + n²) + 47(s + n). The bounds we find are independent of the solvability length of the groups.

We give a topological interpretation of the free metabelian group, following the plan described in [11, 12]. Namely, we represent the free metabelian group with d-generators as an extension of the group of the first homology of the d-dimensional lattice (as Cayley graph of the group ℤ^d), with a canonical 2-cocycle. This construction opens a possibility to study metabelian groups from new points of view; in particular to give useful normal forms of the elements of the group, leading to applications to the random walks, and so on. We also describe the satellite groups which correspond to all 2-cocycles of cohomology group associated with the free solvable groups. The homology of the Cayley graph can be used for studying the wide class of groups which include the class of all solvable groups.

We give a survey of new characterizations of finite solvable groups and the solvable radical of an arbitrary finite group which were obtained over the past decade. We also discuss generalizations of these results to some classes of infinite groups and their analogues for Lie algebras. Some open problems are discussed as well.

Let D be an F-central non-commutative division ring. Here, it is proved that if GL_n(D) contains a non-abelian soluble maximal subgroup, then n = 1, [D : F] < ∞, and D is cyclic of degree p, a prime. Furthermore, a classification of soluble maximal subgroups of GL_n(F) for an algebraically closed or real closed field F is also presented. We then determine all soluble maximal subgroups of GL₂(F) for fields F with Char F ≠ 2.

In this paper, it is shown that a finite group G is always supersolvable if |N_G(H) : H| ≤ 2 for every non-cyclic subgroup H of G of prime-power order. Also, finite groups with all supersolvable non-cyclic subgroups being self-normalizing, and finite p-groups with all non-cyclic proper subgroups being of prime index in their normalizers are completely classified.

Let G be a finite group. A subgroup H of G is called an ℋ-subgroup of G if N_G(H) ∩ H^g ≤ H for all g ∈ G; G is said to be an ℋ_p-group if every cyclic subgroup of G of prime order or order 4 is an ℋ-subgroup of G. In this paper, the structure of the finite groups all of whose maximal subgroups of even order are ℋ_p-subgroups have been characterized.

For a finite group G, let γ(G) denote the number of conjugacy classes of all non-nilpotent subgroups of G, and let π(G) denote the set of the prime divisors of |G|. In this paper, we establish lower bounds on γ(G). In fact, we show that if G is a finite solvable group, then γ(G) = 0 or γ(G) ≥ 2^|π(G)|-2, and if G is non-solvable, then γ(G) ≥ |π(G)| + 1. Both lower bounds are best possible.

In this paper, we introduce the smallest equivalence relation ${\tau }^{\ast }$ on a hypergroup $H$ such that the quotient $H/{\tau }^{\ast }$, the set of all equivalence classes, is a solvable group. The characterization of solvable groups via strongly regular relations is investigated and several results on the topic are presented.

Let $G$ be a finite group and $n$ any divisor of $\left|G\right|$, the order of $G$. Let $F_n(G)=\{x\in G|x^n=1\}$, Frobenius’ theorem states that $|F_n(G)|=f_n\cdot n$ for some positive integer $f_n$. We call $f_n$ a Frobenius quotient of $G$ for $n$. Let ${ℱ}(G)$ be the set of all Frobenius quotients of $G$, we call ${ℱ}(G)$ the Frobenius spectrum of $G$. In this paper, we give a complete classification of finite groups $G$ with ${ℱ}(G)=\{1,p\}$ for $p$ being the smallest prime divisor of $\left|G\right|$. Moreover, let $G$ be a finite group of even order, ${{ℱ}}^{\prime }(G)$ the set of all Frobenius quotients of $G$ for even divisors of $\left|G\right|$ and $\mathrm{\text{mfe}}(G)$ the maximum Frobenius quotient in ${{ℱ}}^{\prime }(G)$, we prove that $G$ is always solvable if $\mathrm{\text{mfe}}(G)\le 7$ or $\mathrm{\text{mfe}}(G)=8$ and $A_5$ is not a composition factor of $G$.

Let $G$ be a finite group. If $K$ is a subgroup of $G$ and $H$ a subgroup of $K$, we say that $H$ is strongly closed in $K$ with respect to $G$ if $K\cap H^g\le H$ for any $g\in G$. We say that a subgroup $H$ of $G$ is strongly closed in $G$ if $H$ is strongly closed in $N_G(H)$ with respect to $G$. A subgroup $H$ of a group $G$ is said to be weakly $c$-supplemented in $G$ if $G$ has a subgroup $K$ such that $G=HK$ and $H\cap K$ is strongly closed in $G$. In this paper, we study the structure of a group $G$ under the assumption, that some subgroups of prime power order are weakly $c$-supplemented in $G$. Our results extend and generalize several recent results in the literature.

Let $G$ be a solvable, nonnilpotent and directly indecomposable finite group. Suppose $G$ has exactly two nonlinear nonfaithful irreducible characters ${\chi }_1,{\chi }_2$. In this paper, we classify the finite groups that satisfy these properties with $\mathrm{\text{Ker}}{\chi }_1\cap \mathrm{\text{Ker}}{\chi }_2>1$.

In this paper, we consider the influence of the number of centralizers of noncentral elements of odd order on the solvability of finite groups. Also, we characterize some finite groups in which the centralizer of every noncentral element of odd order is abelian.

Denote by ${\nu }_p(G)$ the number of Sylow $p$-subgroups of $G$. For every subgroup $H$ of $G$, it is easy to see that ${\nu }_p(H)\le {\nu }_p(G)$, but ${\nu }_p(H)$ does not divide ${\nu }_p(G)$ in general. Following [W. Guo and E. P. Vdovin, Number of Sylow subgroups in finite groups, J. Group Theory 21(4) (2018) 695–712], we say that a group $G$ satisfies DivSyl(p) if ${\nu }_p(H)$ divides ${\nu }_p(G)$ for every subgroup $H$ of $G$. In this paper, we show that “almost for every” finite simple group $S$, there exists a prime $p$ such that $S$ does not satisfy DivSyl(p).

The purpose of this paper is to introduce a concept of commutator in a multiplicative Lie algebra which will be termed as a Lie commutator. Consequently, we develop the theory of Lie solvability and Lie nilpotency of multiplicative Lie algebras.

We characterize the structure of finite unitary rings $R$ in which every proper subring is commutative and show that $R$ is a commutative ring.

Let R be a finite unitary ring whose group of units is not solvable but all groups of units of all its proper subrings are solvable. In this paper, we classify these rings and show that all finite rings of order $p^n$ for $n<5$ and some of order $p^6$ are in this class of rings.

Let G be a finite group and p an odd prime. Let be the set of proper subgroups M of G with |G:M| not a prime power and |G:M|_p=1. In this paper, we investigate the structure of G if every member of is nilpotent. In particular, a new characterization of PSL(2,7) is obtained. The proof of the theorem depends on the classification of finite simple groups.