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
A graph is said to be s-arc-regular if its full automorphism group acts regularly on the set of its s-arcs. In this paper, we investigate connected cubic s-arc-regular Cayley graphs of finite nonabelian simple groups....A graph is said to be s-arc-regular if its full automorphism group acts regularly on the set of its s-arcs. In this paper, we investigate connected cubic s-arc-regular Cayley graphs of finite nonabelian simple groups. Two sufficient and necessary conditions for such graphs to be 1- or 2-arcregular are given and based on the conditions, several infinite families of 1- or 2-arc-regular cubic Cayley graphs of alternating groups are constructed.展开更多
Assume p is an odd prime. We investigate finite p-groups all of whose minimal nonabelian subgroups are of order p^3. Let P1-groups denote the p-groups all of whose minimal nonabelian subgroups are nonme tacyclic of or...Assume p is an odd prime. We investigate finite p-groups all of whose minimal nonabelian subgroups are of order p^3. Let P1-groups denote the p-groups all of whose minimal nonabelian subgroups are nonme tacyclic of order p^3. In this paper, the P1-groups are classified, and as a by-product, we prove the Hughes' conjecture is true for the P1-groups.展开更多
For an odd prime p,we give a criterion for finite p-groups whose nonnormal subgroups are metacyclic,and based on the criterion,the p-groups whose nonnormal subgroups are metacyclic are classified up to isomorphism.Thi...For an odd prime p,we give a criterion for finite p-groups whose nonnormal subgroups are metacyclic,and based on the criterion,the p-groups whose nonnormal subgroups are metacyclic are classified up to isomorphism.This solves a problem proposed by Berkovich.展开更多
A finite group G is said to be a Bn-group if any n-element subset A={a1,a2,...,an}of G satisfies∣∣A^2∣∣=|{aiaj|1≤i,j≤n}|≤n(n+1)/2.In this paper,the characterizations of the B6-and B7-groups are given.
Let G be a finite nonabelian group which has no abelian maximal subgroups and satisfies that any two non-commutative elements generate a maximal subgroup. Then G is isomorphic to the smallest Suzuki 2-group of order 64.
In this paper we investigate the title groups which we call isomaximal. We give the list of all isomaximal 2-groups with abelian maximal subgroups. Further, we prove some properties of isomaximal 2-groups with nonabel...In this paper we investigate the title groups which we call isomaximal. We give the list of all isomaximal 2-groups with abelian maximal subgroups. Further, we prove some properties of isomaximal 2-groups with nonabelian maximal subgroups. After that, we investigate the structure of isomaximal groups of order less than 64. Finally, in Theorem 14. we show that the minimal nonmetacyclic group of order 32 possesses a unique isomaximal extension of order 64.展开更多
A Cayley graph Г=Cay(G,S)is said to be normal if G is normal in Aut Г.In this paper,we investigate the normality problem of the connected 11-valent symmetric Cayley graphs Г of finite nonabelian simple groups G,whe...A Cayley graph Г=Cay(G,S)is said to be normal if G is normal in Aut Г.In this paper,we investigate the normality problem of the connected 11-valent symmetric Cayley graphs Г of finite nonabelian simple groups G,where the vertex stabilizer A_(u) is soluble for A=Aut Г and v∈∨Г.We prove that either Г is normal or G=A_(5),A_(10),A_(54),A_(274),A_(549) or A_(1099).Further,11-valent symmetric nonnormal Cayley graphs of As,A54 and A274 are constructed.This provides some more examples of nonnormal 11-valent symmetric Cayley graphs of finite nonabelian simple groups after the first graph of this kind(of valency 11)was constructed by Fang,Ma and Wang in 2011.展开更多
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 Guangxi Science Foundations (Grant No. 0832054)Guangxi Postgraduate Education Innovation Research (Grant No. 2008105930701M102)
文摘A graph is said to be s-arc-regular if its full automorphism group acts regularly on the set of its s-arcs. In this paper, we investigate connected cubic s-arc-regular Cayley graphs of finite nonabelian simple groups. Two sufficient and necessary conditions for such graphs to be 1- or 2-arcregular are given and based on the conditions, several infinite families of 1- or 2-arc-regular cubic Cayley graphs of alternating groups are constructed.
基金Supported by National Natural Science Foundation of China(Grant Nos.11771258 and 11471198)
文摘Assume p is an odd prime. We investigate finite p-groups all of whose minimal nonabelian subgroups are of order p^3. Let P1-groups denote the p-groups all of whose minimal nonabelian subgroups are nonme tacyclic of order p^3. In this paper, the P1-groups are classified, and as a by-product, we prove the Hughes' conjecture is true for the P1-groups.
基金supported by National Natural Science Foundation of China(Grant Nos.11771258 and 11471198)。
文摘For an odd prime p,we give a criterion for finite p-groups whose nonnormal subgroups are metacyclic,and based on the criterion,the p-groups whose nonnormal subgroups are metacyclic are classified up to isomorphism.This solves a problem proposed by Berkovich.
基金The second author acknowledges the support of the Jiangsu University(Grant No.5501190011).
文摘A finite group G is said to be a Bn-group if any n-element subset A={a1,a2,...,an}of G satisfies∣∣A^2∣∣=|{aiaj|1≤i,j≤n}|≤n(n+1)/2.In this paper,the characterizations of the B6-and B7-groups are given.
基金NSFC (No.10671114)NSF of Shanxi Province (No.20051007)the Returned Overseas(student) Fund of Shanxi province (No.[2007]13-56)
文摘Let G be a finite nonabelian group which has no abelian maximal subgroups and satisfies that any two non-commutative elements generate a maximal subgroup. Then G is isomorphic to the smallest Suzuki 2-group of order 64.
基金supported by Ministry of Science, Education and Sports of Republic of Croatia (Grant No.036-0000000-3223)
文摘In this paper we investigate the title groups which we call isomaximal. We give the list of all isomaximal 2-groups with abelian maximal subgroups. Further, we prove some properties of isomaximal 2-groups with nonabelian maximal subgroups. After that, we investigate the structure of isomaximal groups of order less than 64. Finally, in Theorem 14. we show that the minimal nonmetacyclic group of order 32 possesses a unique isomaximal extension of order 64.
基金supported by the National Natural Science Foundation of China(11701503,11861076,12061089,11761079)Yunnan Applied Basic Research Projects(2018FB003,2019FB139)the third author was supported by the National Natural Science Foundation of China(11601263,11701321).
文摘A Cayley graph Г=Cay(G,S)is said to be normal if G is normal in Aut Г.In this paper,we investigate the normality problem of the connected 11-valent symmetric Cayley graphs Г of finite nonabelian simple groups G,where the vertex stabilizer A_(u) is soluble for A=Aut Г and v∈∨Г.We prove that either Г is normal or G=A_(5),A_(10),A_(54),A_(274),A_(549) or A_(1099).Further,11-valent symmetric nonnormal Cayley graphs of As,A54 and A274 are constructed.This provides some more examples of nonnormal 11-valent symmetric Cayley graphs of finite nonabelian simple groups after the first graph of this kind(of valency 11)was constructed by Fang,Ma and Wang in 2011.
