Let G be a finite abelian group and k be a positive integer.The Davenport constant is a central invariant in zero-sum thoery.The invariant Dk(G)generalizes the Davenport constant D(G)and is defined as the maximum leng...Let G be a finite abelian group and k be a positive integer.The Davenport constant is a central invariant in zero-sum thoery.The invariant Dk(G)generalizes the Davenport constant D(G)and is defined as the maximum length l such that there exists a sequenceB of length l overGcontaining k disjoint non-empty zero-sum subsequences.This paper studies the inverse problem associated with this invariant for the elementary abelian 2-groups C_(2)^(r).For r∈[2,4],we characterize the structures of zero-sum sequences of length D2(C_(2)^(r))and D2(C_(2)^(r))-1 in C_(2)^(r) that can be decomposed into at most two minimal zero-sum subsequences.For r∈[2,5],we characterize the structures of sequences of length D2(C_(2)^(r))-1.展开更多
In order to answer a question motivated by constructing substitution boxes in block ciphers we will exhibit an infinite family of full-rank factorizations of elementary 2-groups into two factors having equal sizes.
The present note determines the structure of the K2-group and of its subgroup over a finite commutative ring R by considering relations between R andfinite commutative local ring Ri (1 < i < m), where R Ri and K...The present note determines the structure of the K2-group and of its subgroup over a finite commutative ring R by considering relations between R andfinite commutative local ring Ri (1 < i < m), where R Ri and K2(R) =K2(Ri). We show that if charKi= p (Ki denotes the residual field of Ri), then K2(Ri) and its subgroups must be p-groups.展开更多
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.
Let p be a prime and F_p be a finite field of p elements.Let F_(pG)denote the group algebra of the finite p-group G over the field F_(p)and V(F_(pG))denote the group of normalized units in F_(pG).Suppose that G and H ...Let p be a prime and F_p be a finite field of p elements.Let F_(pG)denote the group algebra of the finite p-group G over the field F_(p)and V(F_(pG))denote the group of normalized units in F_(pG).Suppose that G and H are finite p-groups given by a central extension of the form 1→Z_(p)^(m)→G→Z_(p)×···×Z_(p)→1 and G'≌Z_(p),m≥1.Then V(F_(p)G)≌V(F_(p)H)if and only if G≌H.Balogh and Bovdi only solved the isomorphism problem when p is odd.In this paper,the case p=2 is determined.展开更多
A group H is said to be autocapable if there exists a group M such that H is isomorphic to the absolute central factor group M/L(M)of M.In this paper,we first prove that if N is a characteristic subgroup of an autocap...A group H is said to be autocapable if there exists a group M such that H is isomorphic to the absolute central factor group M/L(M)of M.In this paper,we first prove that if N is a characteristic subgroup of an autocapable group H,then N is neither the generalized quaternion group nor the semi-dihedral group.Next,we give the classification of finite groups G if G/L(G)is a 2-group of maximal class.展开更多
基金supported by the National Natural Science Foundation of China(No.12301425)。
文摘Let G be a finite abelian group and k be a positive integer.The Davenport constant is a central invariant in zero-sum thoery.The invariant Dk(G)generalizes the Davenport constant D(G)and is defined as the maximum length l such that there exists a sequenceB of length l overGcontaining k disjoint non-empty zero-sum subsequences.This paper studies the inverse problem associated with this invariant for the elementary abelian 2-groups C_(2)^(r).For r∈[2,4],we characterize the structures of zero-sum sequences of length D2(C_(2)^(r))and D2(C_(2)^(r))-1 in C_(2)^(r) that can be decomposed into at most two minimal zero-sum subsequences.For r∈[2,5],we characterize the structures of sequences of length D2(C_(2)^(r))-1.
文摘In order to answer a question motivated by constructing substitution boxes in block ciphers we will exhibit an infinite family of full-rank factorizations of elementary 2-groups into two factors having equal sizes.
文摘The present note determines the structure of the K2-group and of its subgroup over a finite commutative ring R by considering relations between R andfinite commutative local ring Ri (1 < i < m), where R Ri and K2(R) =K2(Ri). We show that if charKi= p (Ki denotes the residual field of Ri), then K2(Ri) and its subgroups must be p-groups.
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National Natural Science Foundation of China(Grant No.12171142)。
文摘Let p be a prime and F_p be a finite field of p elements.Let F_(pG)denote the group algebra of the finite p-group G over the field F_(p)and V(F_(pG))denote the group of normalized units in F_(pG).Suppose that G and H are finite p-groups given by a central extension of the form 1→Z_(p)^(m)→G→Z_(p)×···×Z_(p)→1 and G'≌Z_(p),m≥1.Then V(F_(p)G)≌V(F_(p)H)if and only if G≌H.Balogh and Bovdi only solved the isomorphism problem when p is odd.In this paper,the case p=2 is determined.
基金National Natural Science Foundation of China(12171302,11801334)Natural Science Foundation of Shanxi Province(202103021224287)+1 种基金Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi(2021L278)Science&Technology Development Fund of Tianjin Education Commission for Higher Education(2019KJ141).
文摘A group H is said to be autocapable if there exists a group M such that H is isomorphic to the absolute central factor group M/L(M)of M.In this paper,we first prove that if N is a characteristic subgroup of an autocapable group H,then N is neither the generalized quaternion group nor the semi-dihedral group.Next,we give the classification of finite groups G if G/L(G)is a 2-group of maximal class.
