The following theorem is proved: Let G be a finite group and π e (G) be the set of element orders... more The following theorem is proved: Let G be a finite group and π e (G) be the set of element orders in G. If π e (G) ∩ {2} = ∅; or π e (G) ∩ {3, 4} = ∅; or π e (G) ∩ {3, 5} = ∅, then G is solvable. Moreover, using the intersection with π e (G) being empty set to judge G is solvable or not, only the above three cases. 1 Introduction Let G be a finite group. We have two basic sets: |G| and π e (G). There are many famous works about |G| in the history of group theory. The set π e (G)
Bulletin of the Iranian Mathematical Society, 2020
In this paper, by solving Diophantine equations involving simple $$K_4$$ K 4 -groups, we will try... more In this paper, by solving Diophantine equations involving simple $$K_4$$ K 4 -groups, we will try to point out that it is not easy to prove the infinitude of simple $$K_4$$ K 4 -groups. This problem goes far beyond what is known about Dickson’s conjecture at present.
In this paper, by solving Diophantine equations involving simple K 4-groups, we will try to point... more In this paper, by solving Diophantine equations involving simple K 4-groups, we will try to point out that it is not easy to prove the infinitude of simple K 4-groups. This problem goes far beyond what is known about Dickson's conjecture at present.
We prove that generalized free products of finitely generated free-byfinite or nilpotent-by-finit... more We prove that generalized free products of finitely generated free-byfinite or nilpotent-by-finite groups amalgamating a cyclic subgroup areconjugacy separable. Applying this result we prove a generalization of a conjecture of Fine and Rosenberger [7] that groups of F-type are conjugacy separable.
We give examples to show that unlike generalized free products of groups (g.f.p.) polygonal produ... more We give examples to show that unlike generalized free products of groups (g.f.p.) polygonal products of finitely generated (f.g.) nilpotent groups with cyclic amalgamations need not be residually finite (R ) and polygonal products of finite p-groups with cyclic amalgamations need not be residually nilpotent. However, polygonal products f.g. abelian groups are R , and under certain conditions polygonal products of finite p-groups with cyclic amalgamations are R .
For every group U = P SUn(2 k) with n 5, we find the number of finite groups having the same elem... more For every group U = P SUn(2 k) with n 5, we find the number of finite groups having the same element orders as U .
The goal of this article is to survey new results on the recognition problem. We focus our attent... more The goal of this article is to survey new results on the recognition problem. We focus our attention on the methods recently developed in this area. In each section, we formulate related open problems. In the last two sections, we review arithmetical characterization of spectra of finite simple groups and conclude with a list of groups for which the recognition problem was solved within the last three years.
Search of arithmetical properties that determine a finite group uniquely. Among arithmetical prop... more Search of arithmetical properties that determine a finite group uniquely. Among arithmetical properties the order of the group |G | the set of element orders (the spectrum) ω(G) Problem: Is it true that every finite simple group is uniquely determined by its order and spectrum in class of finite groups? More precisely Question 12.39 in the Kourovka Notebook Is it true that a finite group and a finite simple group are isomorphic if they have the same orders and sets of element orders?
The spectrum of a finite group is the set of element orders of this group. The main goal of this ... more The spectrum of a finite group is the set of element orders of this group. The main goal of this paper is to survey results concerning recognition of finite simple groups by spectrum, in particular, to list all finite simple groups for which the recognition problem is solved.
Bulletin of The Iranian Mathematical Society, 2017
We characterize the finite simple groups $L_2(q)$ by the group orders and the largest element ... more We characterize the finite simple groups $L_2(q)$ by the group orders and the largest element orders, where $q$ is a prime or $q=2^a$, with $2^a+1$ or $2^a-1$ a prime.
In this note we provide some counterexamples for the conjectures of finite simple groups, one of ... more In this note we provide some counterexamples for the conjectures of finite simple groups, one of the conjectures said "all finite simple groups $G$ can be determined using their orders $|G|$ and the number of elements of order $p$, where $p$ the largest prime divisor of $|G|$".
Let G be a finite group, π(G) be the set of prime divisors dividing the order of G and πe(G) (spe... more Let G be a finite group, π(G) be the set of prime divisors dividing the order of G and πe(G) (spectrum) denote the set of element orders of G. We define wo(G) = |π(G)| the width of order of G and ws(G) = max {|π(k)| | k ! πe(G)} the width of spectrum of G. In this paper, we discuss the cases of wo(G) and ws(G) are small, prove several new results and give a survey about the two widths of groups. This article is a part of revised based on the author’s plenary talk at “2020 Ural Workshop on Group Theory and Combinatorics”.
Abdollahi, Akbari, and Maimani put forward a conjecture named AAM's Conjecture as follows. If M i... more Abdollahi, Akbari, and Maimani put forward a conjecture named AAM's Conjecture as follows. If M is a finite nonabelian simple group and G is a group such that () () M G ∇ = ∇ , then M G ≅. In this note, we prove that if G is a finite nonabelian simple group with () () 16
A subgroup H is said to be nc-supplemented in a group G if there exists a subgroup K ≤ G such tha... more A subgroup H is said to be nc-supplemented in a group G if there exists a subgroup K ≤ G such that HK ◁ G and H ∩ K is contained in HG, the core of H in G. We characterize the supersolubility of finite groups G with that every maximal subgroup of the Sylow subgroups is nc-supplemented in G.
The aim of this paper is to show that L 2 (2 f) are the only nonsolvable groups in which every ir... more The aim of this paper is to show that L 2 (2 f) are the only nonsolvable groups in which every irreducible character of even degree vanishes on just one conjugacy class.
Let G be a finite group and π(G) = {p 1 , p 2 , • • • , p s }. For p ∈ π(G), we put deg(p) : = |{... more Let G be a finite group and π(G) = {p 1 , p 2 , • • • , p s }. For p ∈ π(G), we put deg(p) : = |{q ∈ π(G)|p ∼ q in the prime graph of G}|, which is called the degree of p. We also define D(G) := (deg(p 1), deg(p 2),. .. , deg(p s)), where p 1 < p 2 < • • • < p s , which is called the degree pattern of G. We say G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups having the same order and degree pattern as G. In particular, a 1-fold OD-characterizable group is simply called an OD-characterizable group (see [13]). In the present paper, we determine all finite simple groups whose first prime graph components are 1-regular and prove that all finite simple groups whose first prime graph components are r-regular except U 4 (2) are OD-characterizable, where 0 ≤ r ≤ 2. In particular, U 4 (2) is exactly 2-fold ODcharacterizable, which improves a result in [25].
A Conjecture of Abe-Iiyori and Finite Orthogonal Simple Groups with Witt Index n PΩ 2n+(q), III
In this paper the authors have proved the following theorem, which solves a conjecture of S. Abe ... more In this paper the authors have proved the following theorem, which solves a conjecture of S. Abe and N. Iiyori for finite simple groups PΩ+2n(q). Theorem Let G be a finite group and PΩ+2n(q) be one of finite orthogonal simple groups with Witt index n. Then G ∼ = PΩ+2n(q) if and only if Ord(Ssol(G)) = Ord(Ssol(PΩ+2n(q))), where Ord(Ssol(G)) is the set of orders of solvable subgroups in G. Keywords finite orthogonal simple groups, set of orders of solvable subgroups, classification theorem of finite simple groups, solvable prime graph. 2000 MR Subject Classification 20C15 1
The spectrum of a finite group is the set of its element orders. A finite group G is said to be r... more The spectrum of a finite group is the set of its element orders. A finite group G is said to be recognizable by spectrum, if every finite group with the same spectrum as G is isomorphic to G. The purpose of the paper is to prove that for every natural m the finite simple Chevalley group F 4 (2 m) is recognizable by spectrum.
In this article, we prove a conjecture of Thompson for an infinite class of simple groups of Lie ... more In this article, we prove a conjecture of Thompson for an infinite class of simple groups of Lie type E 7 (q). More precisely, we show that every finite group G with the properties Z(G) = 1 and cs(G) = cs(E 7 (q)) is necessarily isomorphic to E 7 (q), where cs(G) and Z(G) are the set of lengths of conjugacy classes of G and the center of G respectively.
Proceedings of the American Mathematical Society, 1993
It is shown that the groups in question are either p-groups of exponent p or Frobenius groups of ... more It is shown that the groups in question are either p-groups of exponent p or Frobenius groups of particular type, or they are isomorphic to the simple group A5 ; the misprints and mistakes of a previous paper of the second author are corrected.
The following theorem is proved: Let G be a finite group and π e (G) be the set of element orders... more The following theorem is proved: Let G be a finite group and π e (G) be the set of element orders in G. If π e (G) ∩ {2} = ∅; or π e (G) ∩ {3, 4} = ∅; or π e (G) ∩ {3, 5} = ∅, then G is solvable. Moreover, using the intersection with π e (G) being empty set to judge G is solvable or not, only the above three cases. 1 Introduction Let G be a finite group. We have two basic sets: |G| and π e (G). There are many famous works about |G| in the history of group theory. The set π e (G)
Bulletin of the Iranian Mathematical Society, 2020
In this paper, by solving Diophantine equations involving simple $$K_4$$ K 4 -groups, we will try... more In this paper, by solving Diophantine equations involving simple $$K_4$$ K 4 -groups, we will try to point out that it is not easy to prove the infinitude of simple $$K_4$$ K 4 -groups. This problem goes far beyond what is known about Dickson’s conjecture at present.
In this paper, by solving Diophantine equations involving simple K 4-groups, we will try to point... more In this paper, by solving Diophantine equations involving simple K 4-groups, we will try to point out that it is not easy to prove the infinitude of simple K 4-groups. This problem goes far beyond what is known about Dickson's conjecture at present.
We prove that generalized free products of finitely generated free-byfinite or nilpotent-by-finit... more We prove that generalized free products of finitely generated free-byfinite or nilpotent-by-finite groups amalgamating a cyclic subgroup areconjugacy separable. Applying this result we prove a generalization of a conjecture of Fine and Rosenberger [7] that groups of F-type are conjugacy separable.
We give examples to show that unlike generalized free products of groups (g.f.p.) polygonal produ... more We give examples to show that unlike generalized free products of groups (g.f.p.) polygonal products of finitely generated (f.g.) nilpotent groups with cyclic amalgamations need not be residually finite (R ) and polygonal products of finite p-groups with cyclic amalgamations need not be residually nilpotent. However, polygonal products f.g. abelian groups are R , and under certain conditions polygonal products of finite p-groups with cyclic amalgamations are R .
For every group U = P SUn(2 k) with n 5, we find the number of finite groups having the same elem... more For every group U = P SUn(2 k) with n 5, we find the number of finite groups having the same element orders as U .
The goal of this article is to survey new results on the recognition problem. We focus our attent... more The goal of this article is to survey new results on the recognition problem. We focus our attention on the methods recently developed in this area. In each section, we formulate related open problems. In the last two sections, we review arithmetical characterization of spectra of finite simple groups and conclude with a list of groups for which the recognition problem was solved within the last three years.
Search of arithmetical properties that determine a finite group uniquely. Among arithmetical prop... more Search of arithmetical properties that determine a finite group uniquely. Among arithmetical properties the order of the group |G | the set of element orders (the spectrum) ω(G) Problem: Is it true that every finite simple group is uniquely determined by its order and spectrum in class of finite groups? More precisely Question 12.39 in the Kourovka Notebook Is it true that a finite group and a finite simple group are isomorphic if they have the same orders and sets of element orders?
The spectrum of a finite group is the set of element orders of this group. The main goal of this ... more The spectrum of a finite group is the set of element orders of this group. The main goal of this paper is to survey results concerning recognition of finite simple groups by spectrum, in particular, to list all finite simple groups for which the recognition problem is solved.
Bulletin of The Iranian Mathematical Society, 2017
We characterize the finite simple groups $L_2(q)$ by the group orders and the largest element ... more We characterize the finite simple groups $L_2(q)$ by the group orders and the largest element orders, where $q$ is a prime or $q=2^a$, with $2^a+1$ or $2^a-1$ a prime.
In this note we provide some counterexamples for the conjectures of finite simple groups, one of ... more In this note we provide some counterexamples for the conjectures of finite simple groups, one of the conjectures said "all finite simple groups $G$ can be determined using their orders $|G|$ and the number of elements of order $p$, where $p$ the largest prime divisor of $|G|$".
Let G be a finite group, π(G) be the set of prime divisors dividing the order of G and πe(G) (spe... more Let G be a finite group, π(G) be the set of prime divisors dividing the order of G and πe(G) (spectrum) denote the set of element orders of G. We define wo(G) = |π(G)| the width of order of G and ws(G) = max {|π(k)| | k ! πe(G)} the width of spectrum of G. In this paper, we discuss the cases of wo(G) and ws(G) are small, prove several new results and give a survey about the two widths of groups. This article is a part of revised based on the author’s plenary talk at “2020 Ural Workshop on Group Theory and Combinatorics”.
Abdollahi, Akbari, and Maimani put forward a conjecture named AAM's Conjecture as follows. If M i... more Abdollahi, Akbari, and Maimani put forward a conjecture named AAM's Conjecture as follows. If M is a finite nonabelian simple group and G is a group such that () () M G ∇ = ∇ , then M G ≅. In this note, we prove that if G is a finite nonabelian simple group with () () 16
A subgroup H is said to be nc-supplemented in a group G if there exists a subgroup K ≤ G such tha... more A subgroup H is said to be nc-supplemented in a group G if there exists a subgroup K ≤ G such that HK ◁ G and H ∩ K is contained in HG, the core of H in G. We characterize the supersolubility of finite groups G with that every maximal subgroup of the Sylow subgroups is nc-supplemented in G.
The aim of this paper is to show that L 2 (2 f) are the only nonsolvable groups in which every ir... more The aim of this paper is to show that L 2 (2 f) are the only nonsolvable groups in which every irreducible character of even degree vanishes on just one conjugacy class.
Let G be a finite group and π(G) = {p 1 , p 2 , • • • , p s }. For p ∈ π(G), we put deg(p) : = |{... more Let G be a finite group and π(G) = {p 1 , p 2 , • • • , p s }. For p ∈ π(G), we put deg(p) : = |{q ∈ π(G)|p ∼ q in the prime graph of G}|, which is called the degree of p. We also define D(G) := (deg(p 1), deg(p 2),. .. , deg(p s)), where p 1 < p 2 < • • • < p s , which is called the degree pattern of G. We say G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups having the same order and degree pattern as G. In particular, a 1-fold OD-characterizable group is simply called an OD-characterizable group (see [13]). In the present paper, we determine all finite simple groups whose first prime graph components are 1-regular and prove that all finite simple groups whose first prime graph components are r-regular except U 4 (2) are OD-characterizable, where 0 ≤ r ≤ 2. In particular, U 4 (2) is exactly 2-fold ODcharacterizable, which improves a result in [25].
A Conjecture of Abe-Iiyori and Finite Orthogonal Simple Groups with Witt Index n PΩ 2n+(q), III
In this paper the authors have proved the following theorem, which solves a conjecture of S. Abe ... more In this paper the authors have proved the following theorem, which solves a conjecture of S. Abe and N. Iiyori for finite simple groups PΩ+2n(q). Theorem Let G be a finite group and PΩ+2n(q) be one of finite orthogonal simple groups with Witt index n. Then G ∼ = PΩ+2n(q) if and only if Ord(Ssol(G)) = Ord(Ssol(PΩ+2n(q))), where Ord(Ssol(G)) is the set of orders of solvable subgroups in G. Keywords finite orthogonal simple groups, set of orders of solvable subgroups, classification theorem of finite simple groups, solvable prime graph. 2000 MR Subject Classification 20C15 1
The spectrum of a finite group is the set of its element orders. A finite group G is said to be r... more The spectrum of a finite group is the set of its element orders. A finite group G is said to be recognizable by spectrum, if every finite group with the same spectrum as G is isomorphic to G. The purpose of the paper is to prove that for every natural m the finite simple Chevalley group F 4 (2 m) is recognizable by spectrum.
In this article, we prove a conjecture of Thompson for an infinite class of simple groups of Lie ... more In this article, we prove a conjecture of Thompson for an infinite class of simple groups of Lie type E 7 (q). More precisely, we show that every finite group G with the properties Z(G) = 1 and cs(G) = cs(E 7 (q)) is necessarily isomorphic to E 7 (q), where cs(G) and Z(G) are the set of lengths of conjugacy classes of G and the center of G respectively.
Proceedings of the American Mathematical Society, 1993
It is shown that the groups in question are either p-groups of exponent p or Frobenius groups of ... more It is shown that the groups in question are either p-groups of exponent p or Frobenius groups of particular type, or they are isomorphic to the simple group A5 ; the misprints and mistakes of a previous paper of the second author are corrected.
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