Narrow Results

For a finite group $G,$ the superpower graph $S(G)$ of $G$ is an undirected simple graph with vertex set $G$ and two vertices are adjacent in $S(G)$ if and only if the order of one divides the order of the other in $G$. The aim of this paper is to provide tight bounds for the vertex connectivity, discuss Hamiltonian-like properties of superpower graph of finite non-Abelian groups having an element of exponent order.

We also give some general results about superpower graphs and their relation to other graphs such as the Gruenberg–Kegel graph.

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.

It is shown that there exists a finitely generated infinite simple group of infinite commutator width, and that the commutator width of a finitely generated infinite boundedly simple group can be arbitrarily large. Besides, such groups can be constructed with decidable word and conjugacy problems.

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, p′ are joined by an edge if there is an element in G of order pp′. In [G. Y. Chen et al., Recognition of the finite almost simple groups PGL₂(q) by their spectrum, Journal of Group Theory, 10 (2007) 71–85], it is proved that PGL(2, p^k), where p is an odd prime and k > 1 is an integer, is recognizable by its spectrum.

It is proved that if p > 19 is a prime number which is not a Mersenne or Fermat prime and Γ(G) = Γ(PGL(2, p)), then G has a unique nonabelian composition factor which is isomorphic to PSL(2, p). In this paper as the main result, we show that if p is an odd prime and k > 1 is an odd integer, then PGL(2, p^k) is uniquely determined by its prime graph and so these groups are characterizable by their prime graphs.

Let M be a finite group and D(M) be the degree pattern of M. Denote by h_OD(M) the number of isomorphism classes of finite groups G with the same order and degree pattern as M. A finite group M is called k-fold OD-characterizable if h_OD(M) = k. Usually, a 1-fold OD-characterizable group is simply called OD-characterizable. The purpose of this article is twofold. First, it provides some information on the structure of a group from its degree pattern. Second, it proves that the projective special linear groups L₄(4), L₄(8), L₄(9), L₄(11), L₄(13), L₄(16), L₄(17) are OD-characterizable.

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 G be a finite group. Based on the prime graph of G, the order of G can be divided into a product of coprime positive integers. These integers are called order components of G and the set of order components is denoted by OC(G). Some non-abelian simple groups are known to be uniquely determined by their order components. In this paper we prove that almost sporadic simple groups, except Aut(J₂) and Aut(McL), and the automorphism group of PSL(2, 2ⁿ) where n=2^s are also uniquely determined by their order components. Also we discuss about the characterizability of Aut(PSL(2, q)). As corollaries of these results, we generalize a conjecture of J. G. Thompson and another conjecture of W. Shi and J. Bi for the groups under consideration.

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge if there is an element in G of order pq.

It is proved that if p > 11 and p ≢ 1 (mod 12), then PSL(2,p) is uniquely determined by its prime graph. Also it is proved that if p > 7 is a prime number and Γ(G) = Γ(PSL(2,p²)), then G ≅ PSL(2,p²) or G ≅ PSL(2,p²).2, the non-split extension of PSL(2,p²) by ℤ₂. In this paper as the main result we determine finite groups G such that Γ(G) = Γ(PSL(2,q)), where q = p^k. As a consequence of our results we prove that if q = p^k, k > 1 is odd and p is an odd prime number, then PSL(2,q) is uniquely determined by its prime graph and so these groups are characterizable by their prime graph.

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, p′ are joined by an edge if G has an element of order pp′. Let D = D_n(2), where n ≥ 3 or ²D_n(2), where n ≥ 15. In this paper we prove that D is quasirecognizable by prime graph, i.e. every finite group G with Γ(G) = Γ(D), has a unique nonabelian composition factor which is isomorphic to D. Finally, we consider the quasirecognition by spectrum for these groups. Specially we prove that if p = 2ⁿ + 1 ≥ 17 is a prime number, then D_p(2) is recognizable by spectrum.

Let G be a nonabelian group. We define the noncommuting graph ∇(G) of G as follows: its vertex set is G\Z(G), the noncentral elements of G, and two distinct vertices x and y of ∇(G) are joined by an edge if and only if x and y do not commute as elements of G, i.e. [x, y] ≠ 1. The finite group L is said to be recognizable by noncommuting graph if, for every finite group G, ∇(G) ≅ ∇ (L) implies G ≅ L. In the present article, it is shown that the noncommuting graph of a group with trivial center can determine its prime graph. From this, the following theorem is derived. If two finite groups with trivial centers have isomorphic noncommuting graphs, then their prime graphs coincide. It is also proved that the projective special unitary groups U₄(4), U₄(8), U₄(9), U₄(11), U₄(13), U₄(16), U₄(17) and the projective special linear groups L₉(2), L₁₆(2) are recognizable by noncommuting graph.

In this paper, we give some sufficient condition on the number of proper solvable subgroups of a group to be nilpotent or solvable. In fact, we show that every group with at most 5 (respectively, 58) proper solvable subgroups is nilpotent (respectively, solvable). Also these bounds cannot be improved.

We study groups having the property that every non-cyclic subgroup contains its centralizer. The structure of nilpotent and supersolvable groups in this class is described. We also classify finite p-groups and finite simple groups with the above defined property.

In this paper, we show that every locally graded group with a finite number k of derived subgroups is nilpotent-by-abelian-by-(finite of order ≤ k!)-by-abelian. Also we give some sufficient conditions on the number of derived subgroups of a locally graded group to be solvable.

Let $G$ be a finite group and the irreducible character degree set of $G$ is contained in $\{1,a,b,c,ab,ac\}$, where $a,b$, and $c$ are distinct integers. We show that one of the following statements holds: $G$ is solvable; $\mathrm{\text{cd}}(G)=\{1,a,b,c\}=\{1,9,10,16\}$; or $\mathrm{\text{cd}}(G)=\{1,a,b,c\}=\{1,q-1,q,q+1\}$ for some prime power $q>3$.

In this paper, we study symmetric graphs of valency seven and order $4p^n$ with $p$ a prime and $n$ an arbitrary positive integer. It is proved that no such graph exists for any prime $p>5$, thus reducing the study to the case $p\le 5$. The result is then used to determine all such graphs of order $4p^2$, and such graphs of order $4p^n$ with ‘$n\ge 3$’ can be inductively investigated similarly.

In [Li and Chen, A new characterization of the simple group $A_1(p^n)$, Sib. Math. J.53(2) (2012) 213–247.], it is proved that the simple group $A_1(p^n)$ is uniquely determined by the set of orders of its maximal abelian subgroups. Also in [Momen and Khosravi, Groups with the same orders of maximal abelian subgroups as $A_2(q)$, Monatsh. Math.174 (2013) 285–303], the authors proved that if $L=A_2(q)$, where $q$ is not a Mersenne prime, then every finite group with the same orders of maximal abelian subgroups as $L$, is isomorphic to $L$ or an extension of $L$ by a subgroup of the outer automorphism group of $L$. In this paper, we prove that if $G$ is a finite group with the same orders of maximal abelian subgroups as $E_6(q)$, then $G$ has a unique nonabelian composition factor which is isomorphic to $E_6(q)$.

A high cited paper [Feng and Li, One-regular graphs of square-free order of prime valency, Europ. J. Combin.32 (2011) 261–275] classified 1-arc-regular prime-valent graphs (necessarily Cayley graphs) of square-free order. It is thus interesting to ask what are the $s$-arc-regular prime-valent Cayley graphs of square-free order for each $s\ge 2$. In this paper, a complete list of all such graphs is given. The proving process also involves a complete list of the vertex stabilizers of $s$-arc-regular prime-valent graphs of any order.

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.

For any group $G$, let $\mathrm{\text{nacent}}(G)$ denote the set of all non-abelian centralizers of $G$. Jafarian Amiri and Rostami, Groups with a few nonabelian centralizers, Publ. Math. Debrecen87(3–4) (2015) 429–437 put forward the following question: Let $H$ and $G$ be finite simple groups. Is it true that if $|\mathrm{\text{nacent}}(H)|=|\mathrm{\text{nacent}}(G)|$, then $G$ is isomorphic to $H$? In this paper, among other things, we give a negative answer to this question.

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).