Mohammad Zarrin - Academia.edu (original) (raw)

Papers by Mohammad Zarrin

International Electronic Journal of Algebra

In this paper we present a new sufficient condition for a solubility criterion in terms of centra... more In this paper we present a new sufficient condition for a solubility criterion in terms of centralizers of elements. This result is a corrigendum of one of Zarrin's results. Furthermore, we extend some of K. Khoramshahi and M. Zarrin's results in the primitive case.

Comptes Rendus Mathematique, 2021

Let G be a finite group. We prove that if the number of Sylow 3-subgroups of G is at most 7 and t... more Let G be a finite group. We prove that if the number of Sylow 3-subgroups of G is at most 7 and the number of Sylow 5-subgroups of G is at most 1455, then G is solvable. This is a strong form of a recent conjecture of Robati. 2020 Mathematics Subject Classification. 20D10, 20D20, 20F16, 20F19. Funding. The first author is supported by both TU Graz (R-1501000001) and partial funding from the Austrian Science Fund (FWF): P30934–N35, F05503, F05510. He is also at the University of Nigeria, Nsukka (UNN). The research of the second author is supported by Ministerio de Ciencia e Innovación PID−2019−103854GB−100, Generalitat Valenciana AICO/2020/298 and FEDER funds. Manuscript received 4th October 2020, revised and accepted 5th November 2020.

International Journal of Group Theory, 2017

Let GGG be a group and mathcalNmathcal{N}mathcalN be the class of all nilpotent groups. A subset AAA of GGG is ... more Let GGG be a group and mathcalNmathcal{N}mathcalN be the class of all nilpotent groups. A subset AAA of GGG is said to be nonnilpotent if for any two distinct elements aaa and bbb in AAA, langlea,branglenotinmathcalNlangle a, brangle notin mathcal{N}langlea,branglenotinmathcalN. If, for any other nonnilpotent subset BBB in GGG, ∣A∣geq∣B∣|A|geq |B|∣A∣geq∣B∣, then AAA is said to be a maximal nonnilpotent subset and the cardinality of this subset (if it exists) is denoted by omega(mathcalNG)omega(mathcal{N}_G)omega(mathcalNG). In this paper, among other results, we obtain omega(mathcalNSuz(q))omega(mathcal{N}_{Suz(q)})omega(mathcalNSuz(q)) and omega(mathcalNPGL(2,q))omega(mathcal{N}_{PGL(2,q)})omega(mathcalNPGL(2,q)), where Suz(q)Suz(q)Suz(q) is the Suzuki simple group over the field with qqq elements and PGL(2,q)PGL(2,q)PGL(2,q) is the projective general linear group of degree 222 over the finite field with qqq elements, respectively.

Journal of Algebra, 2021

Let o(G) be the average order of the elements of G, where G is a finite group. We show that there... more Let o(G) be the average order of the elements of G, where G is a finite group. We show that there is no polynomial lower bound for o(G) in terms of o(N), where N G, even when G is a prime-power order group and N is abelian. This gives a negative answer to a question of A. Jaikin-Zapirain. By the condition p 3/c in this theorem, putting c = 1/2 we obtain a negative answer to Question 1.1 for all primes p 7. But in fact a more careful consideration of the parameters involved gives a negative answer for all primes p 5. Corollary 1.3. Question 1.1 has a negative answer for any prime p 5.

Journal of Algebra and Its Applications, 2019

For any group [Formula: see text], let [Formula: see text] denote the set of all non-abelian cent... more For any group [Formula: see text], let [Formula: see text] denote the set of all non-abelian centralizers of [Formula: see text]. Jafarian Amiri and Rostami, Groups with a few nonabelian centralizers, Publ. Math. Debrecen 87(3–4) (2015) 429–437 put forward the following question: Let [Formula: see text] and [Formula: see text] be finite simple groups. Is it true that if [Formula: see text], then [Formula: see text] is isomorphic to [Formula: see text]? In this paper, among other things, we give a negative answer to this question.

Mathematische Nachrichten, 2018

Let G be a group and T < G. A set Π = {H 1 , H 2 ,. .. , Hn} of proper subgroups of G is said to ... more Let G be a group and T < G. A set Π = {H 1 , H 2 ,. .. , Hn} of proper subgroups of G is said to be a strict T-partition of G, if G = ∪ n i=1 H i and H i ∩ H j = T for every 1 ≤ i, j ≤ n. If Π is a strict T-partition of G and the orders of all components of Π are equal, then we say that G has an ET-partition. Here we show that: A finite group G is nilpotent if and only if every subgroup H of G has an ES-partition, for some S ≤ H.

Czechoslovak Mathematical Journal, 2018

In this paper we show that a finite nonabelian characteristically simple group G satisfying n = |... more In this paper we show that a finite nonabelian characteristically simple group G satisfying n = |π(G)| + 2 if and only if G ∼ = A 5 , where n is the number of isomorphism classes of derived subgroups of G and π(G) is the set of prime divisors of the group G. Also, we give a negative answer to a question raised in [11].

Let G be a group and N be the class of nilpotent groups. A subset A of G is said to be nonnilpote... more Let G be a group and N be the class of nilpotent groups. A subset A of G is said to be nonnilpotent if for any two distinct elements a and b in A, ha, bi 62 N. If, for any other nonnilpotent subset B in G, |A| ? |B|, then A is said to be a maximal nonnilpotent subset and the cardinality of this subset (if it exists) is denoted by !(NG). In this paper, among other results, we obtain !(NSuz(q)) and !(NPGL(2,q)), where Suz(q) is the Suzuki simple group over the field with q elements and PGL(2, q) is the projective general linear group of degree 2 over the finite field of size q, respectively.

Bulletin of the Australian Mathematical Society, 2015

A subset$X$of a group$G$is a set of pairwise noncommuting elements if$ab\neq ba$for any two disti... more A subset$X$of a group$G$is a set of pairwise noncommuting elements if$ab\neq ba$for any two distinct elements$a$and$b$in$X$. If$|X|\geq |Y|$for any other set of pairwise noncommuting elements$Y$in$G$, then$X$is called a maximal subset of pairwise noncommuting elements and the cardinality of such a subset (if it exists) is denoted by${\it\omega}(G)$. In this paper, among other things, we prove that, for each positive integer$n$, there are only finitely many groups$G$, up to isoclinism, with${\it\omega}(G)=n$, and we obtain similar results for groups with exactly$n$centralisers.

Journal of Pure and Applied Algebra, 2013

According to Möhres's Theorem an arbitrary group whose proper subgroups are all subnormal (or a g... more According to Möhres's Theorem an arbitrary group whose proper subgroups are all subnormal (or a group without non-subnormal proper subgroups) is solvable. In this paper we generalize Möhres's Theorem, by proving that every group with at most 56 nonsubnormal subgroups is solvable. Also we show that the derived length of a solvable group with a finite number k of non-n-subnormal subgroups is bounded in terms of n and k.

Communications in Algebra, 2010

We associate a graph N G with a group G (called the non-nilpotent graph of G) as follows: take G ... more We associate a graph N G with a group G (called the non-nilpotent graph of G) as follows: take G as the vertex set and two vertices are adjacent if they generate a non-nilpotent subgroup. In this paper we study the graph theoretical properties of N G and its induced subgraph on G\nil(G), where nil(G) = {x ∈ G | x, y is nilpotent for all y ∈ G}. For any finite group G, we prove that N G has either |Z * (G)| or |Z * (G)| + 1 connected components, where Z * (G) is the hypercenter of G. We give a new characterization for finite nilpotent groups in terms of the non-nilpotent graph. In fact we prove that a finite group G is nilpotent if and only if the set of vertex degrees of N G has at most two elements.

Bulletin of the Australian Mathematical Society, 2012

Groups having exactly one normaliser are well known. They are the Dedekind groups. All finite gro... more Groups having exactly one normaliser are well known. They are the Dedekind groups. All finite groups having exactly two normalisers were classified by Pérez-Ramos [‘Groups with two normalizers’, Arch. Math.50 (1988), 199–203], and Camp-Mora [‘Locally finite groups with two normalizers’, Comm. Algebra28 (2000), 5475–5480] generalised that result to locally finite groups. Then Tota [‘Groups with a finite number of normalizer subgroups’, Comm. Algebra32 (2004), 4667–4674] investigated properties (such as solubility) of arbitrary groups with two, three and four normalisers. In this paper we prove that every finite group with at most 20 normalisers is soluble. Also we characterise all nonabelian simple (not necessarily finite) groups with at most 57 normalisers.

Bulletin of the Australian Mathematical Society, 2006

This paper is an attempt to provide a partial answer to the following question put forward by Ber... more This paper is an attempt to provide a partial answer to the following question put forward by Bernhard H. Neumann in 2000: “Let G be a finite group of order g and assume that however a set M of m elements and a set N of n elements of the group is chosen, at least one element of M commutes with at least one element of N. What relations between g, m, n guarantee that G is Abelian?” We find an exponential function f(m,n) such that every such group G is Abelian whenever |G| > f(m,n) and this function can be taken to be polynomial if G is not soluble. We give an upper bound in terms of m and n for the solubility length of G, if G is soluble.

Algebra Colloquium, 2010

Let G be a non-abelian group. The non-commuting graph [Formula: see text] of G is defined as the ... more Let G be a non-abelian group. The non-commuting graph [Formula: see text] of G is defined as the graph whose vertex set is the non-central elements of G and two vertices are joint if and only if they do not commute. In a finite simple graph Γ, the maximum size of complete subgraphs of Γ is called the clique number of Γ and denoted by ω(Γ). In this paper, we characterize all non-solvable groups G with [Formula: see text], where 57 is the clique number of the non-commuting graph of the projective special linear group PSL (2,7). We also determine [Formula: see text] for all finite minimal simple groups G.

Colloquium Mathematicum, 2015

In this paper, we show that a locally graded group with a finite number m of non-(nilpotent of cl... more In this paper, we show that a locally graded group with a finite number m of non-(nilpotent of class at most n) subgroups is (soluble of class at most [log 2 (n)] + m + 3)-by-(finite of order ≤ m!). Also we show that the derived length of a soluble group with a finite number m of non-(nilpotent of class at most n) subgroups, is at most [log 2 (n)] + m + 1.

Colloquium Mathematicum, 2017

For any group G, let C(G) denote the intersection of the normalizers of centralizers of all eleme... more For any group G, let C(G) denote the intersection of the normalizers of centralizers of all elements of G. Set C 0 = 1. Define C i+1 (G)/C i (G) = C(G/C i (G)) for i ≥ 0. By C∞(G) denote the terminal term of the ascending series. In this paper, we show that a finitely generated group G is nilpotent if and only if G = Cn(G) for some positive integer n.

Bulletin of the Australian Mathematical Society, 2014

In this paper we prove that every group with at most 26 normalisers is soluble. This gives a posi... more In this paper we prove that every group with at most 26 normalisers is soluble. This gives a positive answer to Conjecture 3.6 in the author’s paper [On groups with a finite number of normalisers’, Bull. Aust. Math. Soc.86 (2012), 416–423].