A subset D of a group G is a determining set of G if every automorphism of G is uniquely determined by its action on D,and the determining number of G,a(G),is the cardinality of a smallest determining set.A group G is...A subset D of a group G is a determining set of G if every automorphism of G is uniquely determined by its action on D,and the determining number of G,a(G),is the cardinality of a smallest determining set.A group G is called a DEG-group if α(G)equals(G),the generating number of G.Our main results are as follows.Finite groups with determining number 0 or 1 are classified;finite simple groups and finite nilpotent groups are proved to be DEG-groups;for a given finite group H,there is a DEG-group G such that H is isomorphic to a normal subgroup of G and there is an injective mapping from the set of all finite groups to the set of finite DEG-groups;for any integer k≥2,there exists a group G such that α(G)=2 and(G)≥k.展开更多
Assume that G is a finite non-abelian p-group.If G has an abelian maximal subgroup whose number of Generators is at least n,then G is called an M_(n)-group.For p=2,M_(2)-groups have been classified.For odd prime p,thi...Assume that G is a finite non-abelian p-group.If G has an abelian maximal subgroup whose number of Generators is at least n,then G is called an M_(n)-group.For p=2,M_(2)-groups have been classified.For odd prime p,this paper provides the isomorphism classification of M_(2)-groups,thereby achieving a complete classification of M_(2)-groups.展开更多
In this paper, groups of order p^n in which the number of subgroups of possible order is less than or equal to p3 ale classified. It turns out that if p 〉 2, n≥ 5, then the classification of groups of order p^n in w...In this paper, groups of order p^n in which the number of subgroups of possible order is less than or equal to p3 ale classified. It turns out that if p 〉 2, n≥ 5, then the classification of groups of order p^n in which the number of subgroups of possible order is less than or equal to p3 and the classification of groups of order p^n with a cyclic subgroup of index p2 are the same.展开更多
A finite p-group G is called an At-group if t is the minimal non-negative integer such that all subgroups of index pt of G are abelian.The finite p-groups G with H'=G'for all A2-subgroups H of G are classified...A finite p-group G is called an At-group if t is the minimal non-negative integer such that all subgroups of index pt of G are abelian.The finite p-groups G with H'=G'for all A2-subgroups H of G are classified completely in this paper.As an application,a problem proposed by Berkovich is solved.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 subgroup of index p^k of a finite p-group G is called a k-maximal subgroup of G.Denote by d(G) the number of elements in a minimal generator-system of G and by δ_k(G) the number of k-maximal subgroups which do not ...A subgroup of index p^k of a finite p-group G is called a k-maximal subgroup of G.Denote by d(G) the number of elements in a minimal generator-system of G and by δ_k(G) the number of k-maximal subgroups which do not contain the Frattini subgroup of G.In this paper,the authors classify the finite p-groups with δ_(d(G))(G) ≤ p^2 and δ_(d(G)-1)(G) = 0,respectively.展开更多
基金supported by the National Natural Science Foundation of China(11971474,12371025)supported by the National Natural Science Foundation of China(12271318).
文摘A subset D of a group G is a determining set of G if every automorphism of G is uniquely determined by its action on D,and the determining number of G,a(G),is the cardinality of a smallest determining set.A group G is called a DEG-group if α(G)equals(G),the generating number of G.Our main results are as follows.Finite groups with determining number 0 or 1 are classified;finite simple groups and finite nilpotent groups are proved to be DEG-groups;for a given finite group H,there is a DEG-group G such that H is isomorphic to a normal subgroup of G and there is an injective mapping from the set of all finite groups to the set of finite DEG-groups;for any integer k≥2,there exists a group G such that α(G)=2 and(G)≥k.
文摘Assume that G is a finite non-abelian p-group.If G has an abelian maximal subgroup whose number of Generators is at least n,then G is called an M_(n)-group.For p=2,M_(2)-groups have been classified.For odd prime p,this paper provides the isomorphism classification of M_(2)-groups,thereby achieving a complete classification of M_(2)-groups.
基金supported by the National Natural Science Foundation of China(No.10671114)the ShanxiProvincial Natural Science Foundation of China(No.2008012001)the Returned Abroad-StudentFund of Shanxi Province(No.[2007]13-56)
文摘In this paper, groups of order p^n in which the number of subgroups of possible order is less than or equal to p3 ale classified. It turns out that if p 〉 2, n≥ 5, then the classification of groups of order p^n in which the number of subgroups of possible order is less than or equal to p3 and the classification of groups of order p^n with a cyclic subgroup of index p2 are the same.
基金supported by the National Natural Science Foundation of China(nos.12171213,11771191,11771258).
文摘A finite p-group G is called an At-group if t is the minimal non-negative integer such that all subgroups of index pt of G are abelian.The finite p-groups G with H'=G'for all A2-subgroups H of G are classified completely in this paper.As an application,a problem proposed by Berkovich is solved.
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supported by the National Natural Science Foundation of China(Nos.11371232,11371177)
文摘A subgroup of index p^k of a finite p-group G is called a k-maximal subgroup of G.Denote by d(G) the number of elements in a minimal generator-system of G and by δ_k(G) the number of k-maximal subgroups which do not contain the Frattini subgroup of G.In this paper,the authors classify the finite p-groups with δ_(d(G))(G) ≤ p^2 and δ_(d(G)-1)(G) = 0,respectively.
