Let G be a finite group and π(G) = {pl,p2,…… ,pk} be the set of the primes dividing the order of G. We define its prime graph F(G) as follows. The vertex set of this graph is 7r(G), and two distinct vertices ...Let G be a finite group and π(G) = {pl,p2,…… ,pk} be the set of the primes dividing the order of G. We define its prime graph F(G) as follows. The vertex set of this graph is 7r(G), and two distinct vertices p, q are joined by an edge if and only if pq ∈ πe(G). In this case, we write p - q. For p ∈π(G), put deg(p) := |{q ∈ π(G)|p - q}|, which is called the degree of p. We also define D(G) := (deg(p1), deg(p2),..., deg(pk)), where pl 〈 p2 〈 -……〈 pk, which is called the degree pattern of G. We say a group G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups with the same order and degree pattern as G. Specially, a l-fold OD-characterizable group is simply called an OD-characterizable group. Let L := U6(2). In this article, we classify all finite groups with the same order and degree pattern as an almost simple groups related to L. In fact, we prove that L and L.2 are OD-characterizable, L.3 is 3-fold OD-characterizable, and L.S3 is 5-fold OD-characterizable.展开更多
In this paper, we prove that if p, q are distinct primes, (p,q)≡(1,7) (mod 12) and Legendres symbol pq=1 , then the equation 1+p a=2 bq c+2 dp eq f has only solutions of the form (a,b,c,d,e,f)=...In this paper, we prove that if p, q are distinct primes, (p,q)≡(1,7) (mod 12) and Legendres symbol pq=1 , then the equation 1+p a=2 bq c+2 dp eq f has only solutions of the form (a,b,c,d,e,f)=(t,0,0,0,t,0), where t is a non negative integer. We also give all solutions of a kind of generalized Ramanujan Nagell equations by using the theories of imaginary quadratic field and Pells equation.展开更多
Let G be a finite group and Irr(G)the set of all irreducible complex characters of G.Let cd(G)be the set of all irreducible complex character degrees of G and denote byρ(G)the set of all primes which divide some char...Let G be a finite group and Irr(G)the set of all irreducible complex characters of G.Let cd(G)be the set of all irreducible complex character degrees of G and denote byρ(G)the set of all primes which divide some character degree of G.The character-prime graphΓ(G)associated to G is a simple undirected graph whose vertex set isρ(G)and there is an edge between two distinct primes p and q if and only if the product p q divides some character degree of G.We show that the finite nonabelian simple groups A_(7),J_(1),J_(3),J_(4),L_(3)(3)and U_(3)(4)are uniquely determined by their degree-patterns and orders.展开更多
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 si...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.展开更多
Let G be a finite group with order |G|=p1^α1p2^α2……pk^αk, where p1 〈 p2 〈……〈 Pk are prime numbers. One of the well-known simple graphs associated with G is the prime graph (or Gruenberg- Kegel graph) den...Let G be a finite group with order |G|=p1^α1p2^α2……pk^αk, where p1 〈 p2 〈……〈 Pk are prime numbers. One of the well-known simple graphs associated with G is the prime graph (or Gruenberg- Kegel graph) denoted .by г(G) (or GK(G)). This graph is constructed as follows: The vertex set of it is π(G) = {p1,p2,…,pk} and two vertices pi, pj with i≠j are adjacent by an edge (and we write pi - pj) if and only if G contains an element of order pipj. The degree deg(pi) of a vertex pj ∈π(G) is the number of edges incident on pi. We define D(G) := (deg(p1), deg(p2),..., deg(pk)), which is called the degree pattern of G. A group G is called k-fold OD-characterizable if there exist exactly k non- isomorphic groups H such that |H| = |G| and D(H) = D(G). Moreover, a 1-fold OD-characterizable group is simply called OD-characterizable. Let L := U3(5) be the projective special unitary group. In this paper, we classify groups with the same order and degree pattern as an almost simple group related to L. In fact, we obtain that L and L.2 are OD-characterizable; L.3 is 3-fold OD-characterizable; L.S3 is 6-fold OD-characterizable.展开更多
For G a finite group,π_e(G)denotes the set of orders of elements in G.If Ω is a subset of the set of natural numbers,h(Ω)stands for the number of isomorphism classes of finite groups with the same set Ω of element...For G a finite group,π_e(G)denotes the set of orders of elements in G.If Ω is a subset of the set of natural numbers,h(Ω)stands for the number of isomorphism classes of finite groups with the same set Ω of element orders.We say that G is k-distinguishable if h(π_(G))=k<∞,otherwise G is called non-distinguishable.Usually,a 1-distinguishable group is called a characterizable group.It is shown that if M is a sporadic simple group different from M_(12),M_(22),J_2,He,Suz,M^cL and O'N, then Aut(M)is charaeterizable by its dement orders.It is also proved that if M is isomorphic to M_(12),M_(22),He,Suz or O'N,then h(π_e(Aut(M)))∈{1,∞}.展开更多
In the framework of the SU(3) simple group model, we consider the single top quark production process e^-γ →vebt^-. We find that the correction effects on the process mainly come from the terms of the tree-level ...In the framework of the SU(3) simple group model, we consider the single top quark production process e^-γ →vebt^-. We find that the correction effects on the process mainly come from the terms of the tree-level Wqq' couplings. In the reasonable parameter space of the SU(3) simple group model, the deviation of the total production cross section σ^tot from its SM value is larger than 5%, which might be detected in the future high energy linear e^+e^- collider (LC) experiments.展开更多
Let G be an extension of a finite characteristically simple group by an abelian group or a finite simple group.It is shown that every Coleman automorphism of G is an inner automorphism.Interest in such automorphisms a...Let G be an extension of a finite characteristically simple group by an abelian group or a finite simple group.It is shown that every Coleman automorphism of G is an inner automorphism.Interest in such automorphisms arises from the study of the normalizer problem for integral group rings.展开更多
The spectrum of a finite group is the set of its element orders,and two groups are said to be isospectral if they have the same spectra.A finite group G is said to be recognizable by spectrum,if every finite group iso...The spectrum of a finite group is the set of its element orders,and two groups are said to be isospectral if they have the same spectra.A finite group G is said to be recognizable by spectrum,if every finite group isospectral with G is isomorphic to G.We prove that if S is one of the sporadic simple groups M^(c)L,M_(12),M_(22),He,Suz and O'N,then Aut(S)is recognizable by spectrum.This finishes the proof of the recognizability by spectrum of the automorphism groups of all sporadic simple groups,except J_(2).展开更多
For the non-Abelian simple groups with Abelian Sylow 2-subgroups J. H. Walterhas proved the following famous theorem. Lemma 1. If F is a non-Ablian simple group with Abelian Sylow 2-subgroups, thenone of the following...For the non-Abelian simple groups with Abelian Sylow 2-subgroups J. H. Walterhas proved the following famous theorem. Lemma 1. If F is a non-Ablian simple group with Abelian Sylow 2-subgroups, thenone of the following holds:(i)F≌PSL(2,Q),q】3,q≡3,5(mod 8) or q=2<sup>n</sup>,n≥2;(ii)F≌J<sub>1</sub>;(iii)F≌R(q),q=3<sup>2m+1</sup>,m≥1.Let G be a finite group and let π<sub>e</sub>(G) denote the set of all orders of elements展开更多
In this paper, we obtain some classification theorems of finite simple groups with two subgroups of coprime indices which are both supersolvable or one supersolvable and the other nilpotent. Using these classification...In this paper, we obtain some classification theorems of finite simple groups with two subgroups of coprime indices which are both supersolvable or one supersolvable and the other nilpotent. Using these classification theorems, we prove some sufficient conditions of finite solvable groups. Finally, we provide a supplement of Doerks Theorem.展开更多
For a characterχof a finite group G,the number cod(χ) := ∣G:ker(χ)∣/χ(1)is called the codegree ofχ.In this paper,we give a solvability criterion for a finite group G depending on the minimum of the ratioχ(1)^(...For a characterχof a finite group G,the number cod(χ) := ∣G:ker(χ)∣/χ(1)is called the codegree ofχ.In this paper,we give a solvability criterion for a finite group G depending on the minimum of the ratioχ(1)^(2)/cod(χ),whenχvaries among the irreducible characters of G.展开更多
The characterization of A 5 is obtained through the method of calculation.The main result is described as the following: 1)The order of A 5 is one,two,three or five. 2)The element of A 5 is divided into...The characterization of A 5 is obtained through the method of calculation.The main result is described as the following: 1)The order of A 5 is one,two,three or five. 2)The element of A 5 is divided into five conjugate classes. 3)There are fifty and nine subgroup in A 5 and we can obtain one produce element in every subgroup. 4)There are nine conjugate classes in the subgroup of A 5 .展开更多
It is a well-known fact that characters of a finite group can give important information about the group's structure. Also it was proved by the third author of this article that a finite simple group can be uniquely ...It is a well-known fact that characters of a finite group can give important information about the group's structure. Also it was proved by the third author of this article that a finite simple group can be uniquely determined by its character table. Here the authors attempt to investigate how to characterize a finite almost simple group by using less information of its character table, and successfully characterize the almost simple K3-groups by their orders and at most three irreducible character degrees of their character tables.展开更多
We aim to study maximal pairwise commuting sets of 3-transpositions(transvections)of the simple unitary group U_(n)(2)over GF(4),and to construct designs from these sets.Any maximal set of pairwise commuting 3-transpo...We aim to study maximal pairwise commuting sets of 3-transpositions(transvections)of the simple unitary group U_(n)(2)over GF(4),and to construct designs from these sets.Any maximal set of pairwise commuting 3-transpositions is called a basic set of transpositions.Let G=U_(n)(2).It is well known that G is a 3-transposition group with the set D,the conjugacy class consisting of its transvections,as the set of 3-transpositions.Let L be a set of basic transpositions in D.We give general descriptions of L and 1-(ν,κ,λ)designs D=(P,B),with P=D and B={L^(9)|g∈G}.The parameters k=|L|,λ and further properties of D are determined.We also,as examples,apply the method to the unitary simple groups U_(4)(2),U_(5)(2),U_(6)(2),U_(7)(2),U_(8)(2)and U_(9)(2).展开更多
Let G be a finite group. If |N<sub>G</sub>(R<sub>1</sub>)|=|N<sub>L<sub>n</sub></sub>(q)(R<sub>2</sub>)| for every prime r, where R<sub>1</sub...Let G be a finite group. If |N<sub>G</sub>(R<sub>1</sub>)|=|N<sub>L<sub>n</sub></sub>(q)(R<sub>2</sub>)| for every prime r, where R<sub>1</sub>∈Syl<sub>r</sub> G and R<sub>2</sub>∈Syl<sub>r</sub>(L<sub>n</sub>(q)), then G≌L<sub>n</sub>(q).展开更多
For a finite group G, it is denoted by N(G) the set of conjugacy class sizes of G. In 1980s, J. G. Thompson posed the following conjecture: if L is a finite nonabelian simple group, G is a finite group with trivial...For a finite group G, it is denoted by N(G) the set of conjugacy class sizes of G. In 1980s, J. G. Thompson posed the following conjecture: if L is a finite nonabelian simple group, G is a finite group with trivial center, and N(G) = N(L), then L and G are isomorphic. In this paper, it is proved that Thompson's conjecture is true for the alternating group A22 with connected prime graph.展开更多
基金supported by Natural Science Foundation Project of CQ CSTC (2010BB9206)NNSF of China (10871032)+1 种基金Fundamental Research Funds for the Central Universities (Chongqing University, CDJZR10100009)National Science Foundation for Distinguished Young Scholars of China (11001226)
文摘Let G be a finite group and π(G) = {pl,p2,…… ,pk} be the set of the primes dividing the order of G. We define its prime graph F(G) as follows. The vertex set of this graph is 7r(G), and two distinct vertices p, q are joined by an edge if and only if pq ∈ πe(G). In this case, we write p - q. For p ∈π(G), put deg(p) := |{q ∈ π(G)|p - q}|, which is called the degree of p. We also define D(G) := (deg(p1), deg(p2),..., deg(pk)), where pl 〈 p2 〈 -……〈 pk, which is called the degree pattern of G. We say a group G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups with the same order and degree pattern as G. Specially, a l-fold OD-characterizable group is simply called an OD-characterizable group. Let L := U6(2). In this article, we classify all finite groups with the same order and degree pattern as an almost simple groups related to L. In fact, we prove that L and L.2 are OD-characterizable, L.3 is 3-fold OD-characterizable, and L.S3 is 5-fold OD-characterizable.
文摘In this paper, we prove that if p, q are distinct primes, (p,q)≡(1,7) (mod 12) and Legendres symbol pq=1 , then the equation 1+p a=2 bq c+2 dp eq f has only solutions of the form (a,b,c,d,e,f)=(t,0,0,0,t,0), where t is a non negative integer. We also give all solutions of a kind of generalized Ramanujan Nagell equations by using the theories of imaginary quadratic field and Pells equation.
基金supported by NSFC(12071484)Hunan Provincial Natural Science Foundation(2020JJ4675)Foundation of Guangdong University of Science and Technology.
文摘Let G be a finite group and Irr(G)the set of all irreducible complex characters of G.Let cd(G)be the set of all irreducible complex character degrees of G and denote byρ(G)the set of all primes which divide some character degree of G.The character-prime graphΓ(G)associated to G is a simple undirected graph whose vertex set isρ(G)and there is an edge between two distinct primes p and q if and only if the product p q divides some character degree of G.We show that the finite nonabelian simple groups A_(7),J_(1),J_(3),J_(4),L_(3)(3)and U_(3)(4)are uniquely determined by their degree-patterns and orders.
基金supported by Foreign Experts program in Jiangsu Province(No.JSB2018014)supported by the National Natural Science Foundation of China(No.12171126)+1 种基金supported by the RFBR(No.20-51-00007)supported by the National Natural Science Foundation of China(11171364,11671063).
文摘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.
基金Supported by National Natural Science Foundation of China (Grant No. 10871032)the SRFDP of China (Grant No. 20660285002)a subproject of National Natural Science Foundation of China (Grant No. 50674008) (Chongqing University, Nos. 104207520080834, 104207520080968)
文摘Let G be a finite group with order |G|=p1^α1p2^α2……pk^αk, where p1 〈 p2 〈……〈 Pk are prime numbers. One of the well-known simple graphs associated with G is the prime graph (or Gruenberg- Kegel graph) denoted .by г(G) (or GK(G)). This graph is constructed as follows: The vertex set of it is π(G) = {p1,p2,…,pk} and two vertices pi, pj with i≠j are adjacent by an edge (and we write pi - pj) if and only if G contains an element of order pipj. The degree deg(pi) of a vertex pj ∈π(G) is the number of edges incident on pi. We define D(G) := (deg(p1), deg(p2),..., deg(pk)), which is called the degree pattern of G. A group G is called k-fold OD-characterizable if there exist exactly k non- isomorphic groups H such that |H| = |G| and D(H) = D(G). Moreover, a 1-fold OD-characterizable group is simply called OD-characterizable. Let L := U3(5) be the projective special unitary group. In this paper, we classify groups with the same order and degree pattern as an almost simple group related to L. In fact, we obtain that L and L.2 are OD-characterizable; L.3 is 3-fold OD-characterizable; L.S3 is 6-fold OD-characterizable.
基金This work has been partially sopported by the Research Institute for Fundamental Sciences Tabriz,Iran
文摘For G a finite group,π_e(G)denotes the set of orders of elements in G.If Ω is a subset of the set of natural numbers,h(Ω)stands for the number of isomorphism classes of finite groups with the same set Ω of element orders.We say that G is k-distinguishable if h(π_(G))=k<∞,otherwise G is called non-distinguishable.Usually,a 1-distinguishable group is called a characterizable group.It is shown that if M is a sporadic simple group different from M_(12),M_(22),J_2,He,Suz,M^cL and O'N, then Aut(M)is charaeterizable by its dement orders.It is also proved that if M is isomorphic to M_(12),M_(22),He,Suz or O'N,then h(π_e(Aut(M)))∈{1,∞}.
基金Supported by National Natural Science Foundation of China (10775039)Henan Institute of Science and Technology(06040)
文摘In the framework of the SU(3) simple group model, we consider the single top quark production process e^-γ →vebt^-. We find that the correction effects on the process mainly come from the terms of the tree-level Wqq' couplings. In the reasonable parameter space of the SU(3) simple group model, the deviation of the total production cross section σ^tot from its SM value is larger than 5%, which might be detected in the future high energy linear e^+e^- collider (LC) experiments.
基金Project supported by the National Natural Science Foundation(Grant No.10171074)Jiangsu Natural Science Foundation(Grant No.BK200133)the Foundation of State Education Ministry of China
文摘In this paper the following theorem is proved: Every group L3(q) for q = 3^(2m-1)(m≥2) is characterized by its set of element orders.
基金supported by the National Natural Science Foundation of China(Grant No.11871292).
文摘Let G be an extension of a finite characteristically simple group by an abelian group or a finite simple group.It is shown that every Coleman automorphism of G is an inner automorphism.Interest in such automorphisms arises from the study of the normalizer problem for integral group rings.
基金This work is supported by Russian Science Foundation(Project No.14-21-00065).
文摘The spectrum of a finite group is the set of its element orders,and two groups are said to be isospectral if they have the same spectra.A finite group G is said to be recognizable by spectrum,if every finite group isospectral with G is isomorphic to G.We prove that if S is one of the sporadic simple groups M^(c)L,M_(12),M_(22),He,Suz and O'N,then Aut(S)is recognizable by spectrum.This finishes the proof of the recognizability by spectrum of the automorphism groups of all sporadic simple groups,except J_(2).
文摘For the non-Abelian simple groups with Abelian Sylow 2-subgroups J. H. Walterhas proved the following famous theorem. Lemma 1. If F is a non-Ablian simple group with Abelian Sylow 2-subgroups, thenone of the following holds:(i)F≌PSL(2,Q),q】3,q≡3,5(mod 8) or q=2<sup>n</sup>,n≥2;(ii)F≌J<sub>1</sub>;(iii)F≌R(q),q=3<sup>2m+1</sup>,m≥1.Let G be a finite group and let π<sub>e</sub>(G) denote the set of all orders of elements
文摘In this paper, we obtain some classification theorems of finite simple groups with two subgroups of coprime indices which are both supersolvable or one supersolvable and the other nilpotent. Using these classification theorems, we prove some sufficient conditions of finite solvable groups. Finally, we provide a supplement of Doerks Theorem.
基金Supported by the National Natural Science Foundation of China(Grant Nos.11971391,12071376,12301018,12171058,12326356)the Natural Science Foundation of Jiangsu Province(Grant No.BK20231356)+1 种基金the Natural Science Foundation for the Universities in Jiangsu Province(Grant No.23KJB110002)The first and second authors are supported by the Chinese Scholarship Council。
文摘For a characterχof a finite group G,the number cod(χ) := ∣G:ker(χ)∣/χ(1)is called the codegree ofχ.In this paper,we give a solvability criterion for a finite group G depending on the minimum of the ratioχ(1)^(2)/cod(χ),whenχvaries among the irreducible characters of G.
文摘The characterization of A 5 is obtained through the method of calculation.The main result is described as the following: 1)The order of A 5 is one,two,three or five. 2)The element of A 5 is divided into five conjugate classes. 3)There are fifty and nine subgroup in A 5 and we can obtain one produce element in every subgroup. 4)There are nine conjugate classes in the subgroup of A 5 .
基金Supported by Natural Science Foundation of China(Grant Nos.11171364,11271301,11471266 and11426182)"the Fundamental Research Funds for the Central Universities"(Grant Nos.XDJK2014C163,XDJK2014C162)+2 种基金Natural Science Foundation Project of CQ CSTC(Grant No.cstc2014jcyj A00010)Postdoctoral Science Foundation of Chongqing(Grant No.Xm2014029)China Postdoctoral Science Foundation(Grant No.2014M562264)
文摘It is a well-known fact that characters of a finite group can give important information about the group's structure. Also it was proved by the third author of this article that a finite simple group can be uniquely determined by its character table. Here the authors attempt to investigate how to characterize a finite almost simple group by using less information of its character table, and successfully characterize the almost simple K3-groups by their orders and at most three irreducible character degrees of their character tables.
文摘We aim to study maximal pairwise commuting sets of 3-transpositions(transvections)of the simple unitary group U_(n)(2)over GF(4),and to construct designs from these sets.Any maximal set of pairwise commuting 3-transpositions is called a basic set of transpositions.Let G=U_(n)(2).It is well known that G is a 3-transposition group with the set D,the conjugacy class consisting of its transvections,as the set of 3-transpositions.Let L be a set of basic transpositions in D.We give general descriptions of L and 1-(ν,κ,λ)designs D=(P,B),with P=D and B={L^(9)|g∈G}.The parameters k=|L|,λ and further properties of D are determined.We also,as examples,apply the method to the unitary simple groups U_(4)(2),U_(5)(2),U_(6)(2),U_(7)(2),U_(8)(2)and U_(9)(2).
文摘Let G be a finite group. If |N<sub>G</sub>(R<sub>1</sub>)|=|N<sub>L<sub>n</sub></sub>(q)(R<sub>2</sub>)| for every prime r, where R<sub>1</sub>∈Syl<sub>r</sub> G and R<sub>2</sub>∈Syl<sub>r</sub>(L<sub>n</sub>(q)), then G≌L<sub>n</sub>(q).
文摘For a finite group G, it is denoted by N(G) the set of conjugacy class sizes of G. In 1980s, J. G. Thompson posed the following conjecture: if L is a finite nonabelian simple group, G is a finite group with trivial center, and N(G) = N(L), then L and G are isomorphic. In this paper, it is proved that Thompson's conjecture is true for the alternating group A22 with connected prime graph.