In this note, we provide an effective proof of the fundamental structure theorem of finitely generated modules over a principal ideal domain, from which we find the minimality of decomposition for a finitely generated...In this note, we provide an effective proof of the fundamental structure theorem of finitely generated modules over a principal ideal domain, from which we find the minimality of decomposition for a finitely generated module over a principal ideal domain.展开更多
The rational canonical form theorem is very essential basic result of matrix theory, which has been proved by different methods in the literature. In this note, we provide an effcient direct proof, from which the mini...The rational canonical form theorem is very essential basic result of matrix theory, which has been proved by different methods in the literature. In this note, we provide an effcient direct proof, from which the minimality for the decomposition of the rational canonical form can be found.展开更多
It is well known that the Cayley-Hamilton theorem is an interesting and important theorem in linear algebras, which was first explicitly stated by A. Cayley and W. R. Hamilton about in 1858, but the first general proo...It is well known that the Cayley-Hamilton theorem is an interesting and important theorem in linear algebras, which was first explicitly stated by A. Cayley and W. R. Hamilton about in 1858, but the first general proof was published in 1878 by G. Frobenius, and numerous others have appeared since then, for example see [1,2]. From the structure theorem for finitely generated modules over a principal ideal domain it straightforwardly follows the Cayley-Hamilton theorem and the proposition that there exists a vector v in a finite dimensional linear space V such that and a linear transformation of V have the same minimal polynomial. In this note, we provide alternative proofs of these results by only utilizing the knowledge of linear algebras.展开更多
For a nilpotent group G without π-torsion,and x,y ∈ G,if x^(n)=y^(n) for a T-number n,then x=y;if x^(m)y^(n)=y^(n)x^(m) for n-numbers m,n,then xy=yx.This is a wellknown result in group theory.In this paper,we prove ...For a nilpotent group G without π-torsion,and x,y ∈ G,if x^(n)=y^(n) for a T-number n,then x=y;if x^(m)y^(n)=y^(n)x^(m) for n-numbers m,n,then xy=yx.This is a wellknown result in group theory.In this paper,we prove two analogous theorems on matrices,which have independence significance.Specifically,let m be a given positive integer and A a complex square matrix satisfying that(i)all eigenvalues of A are nonnegative,and(i)rank A^(2)=rank A;then A has a unique m-th root X with rank X^(2)=rank X,all eigenvalues of X are nonnegative,and moreover there is a polynomial f(λ)with X=f(A).In addition,let A and B be complex n×n matrices with all eigenvalues nonnegative,and rank A^(2)=rank A,rank B^(2)=rank B;then(i)A=B when A^(r)=B^(r) for some positive integer r,and(i)AB=BA when A^(s)B^(t)=B^(t)A^(s) for two positive integers s and t.展开更多
The automorphism group of G is determined, where G is a nonabelian p-group given by a central extension as 1→Zpm→G→Zp×…×Zp→1 such that its derived subgroup has order p.
Let A be a completely decomposable homogeneous torsion-free abelian group of rank n(n≥2).Let m(n)=A×(a)be the split extension of A by an automorphismαwhich is a cyclic permutation of the direct components twist...Let A be a completely decomposable homogeneous torsion-free abelian group of rank n(n≥2).Let m(n)=A×(a)be the split extension of A by an automorphismαwhich is a cyclic permutation of the direct components twisted by a rational integer m.Then Om(n)is an infinite soluble group.In this paper,the residual finiteness of Om(n)is investigated.展开更多
Let Z>be a principal ideal domain(PID)and M be a module over D.We prove the following two dual results:(i)If M is finitely generated and rr,y are two elements in M such that M/Dx≌M/Dy,then there exists an auto mor...Let Z>be a principal ideal domain(PID)and M be a module over D.We prove the following two dual results:(i)If M is finitely generated and rr,y are two elements in M such that M/Dx≌M/Dy,then there exists an auto morphism a of M such that a(x)=y.(ii)If M satisfies the minimal cond计ion on submodules and X,Y are two locally cyclic submodules of M such that M/X≌M/Y and X≌Y,then there exists an automorphism a of M such that α(X)=Y.展开更多
Let p be a prime and F be a finite field of characteristic p.Suppose that FG is the group algebra of the finite p-group G over the field F.Let V(FG)denote the group of normalized units in FG and let V_(*)(FG)denote th...Let p be a prime and F be a finite field of characteristic p.Suppose that FG is the group algebra of the finite p-group G over the field F.Let V(FG)denote the group of normalized units in FG and let V_(*)(FG)denote the unitary subgroup of V(FG).If p is odd,then the order of V_(*)(FG)is|F|^((|G-1)/2).However,the case p=2 still is open.In this paper,the order of V*(FG)is computed when G is a nonabelian 2-group given by a central extension of the form 1→Z_(2^(n))×Z_(2^(m))→G→Z_(2)×…×Z_(2)→1 and G'≌Z_(2),n,m≥1.Furthermore,a conjecture is confirmed,i.e.,the order of V_(*)(FG)can be divisible by|F|^(1/2(|G|+|Ω1(G)|)-1),where Ω_(1)(G)={g∈G|g^(2)=1}.展开更多
Let G be a group. A subset X of G is said to be non-commuting if xy ≠ yx for any x, y ∈ X with x ≠ y. If {X}≥ IYI for any other non-commuting set Y in G, then X is said to be a maximal non-commuting set. In this p...Let G be a group. A subset X of G is said to be non-commuting if xy ≠ yx for any x, y ∈ X with x ≠ y. If {X}≥ IYI for any other non-commuting set Y in G, then X is said to be a maximal non-commuting set. In this paper, the bound for the cardinality of a maximal non-commuting set in a finite p-group G is determined, where G is a non-abelian p-group given by a central extension as1 → Zp→ G →Zp ×→ × Zp →1 and its derivedsubgroup has order p.展开更多
Let G be a group,and let a be a regular automorphism of order p2 of G,where p is a prime.If G is polycyclic-by-finite and the map φ:G→G defined by=g^φ,[g,a]is surjective,then G is soluble.If G is polycyclic,then CG...Let G be a group,and let a be a regular automorphism of order p2 of G,where p is a prime.If G is polycyclic-by-finite and the map φ:G→G defined by=g^φ,[g,a]is surjective,then G is soluble.If G is polycyclic,then CG(a^p)and G/[G,a^p]are both nilpotent-by-finite.展开更多
文摘In this note, we provide an effective proof of the fundamental structure theorem of finitely generated modules over a principal ideal domain, from which we find the minimality of decomposition for a finitely generated module over a principal ideal domain.
文摘The rational canonical form theorem is very essential basic result of matrix theory, which has been proved by different methods in the literature. In this note, we provide an effcient direct proof, from which the minimality for the decomposition of the rational canonical form can be found.
文摘It is well known that the Cayley-Hamilton theorem is an interesting and important theorem in linear algebras, which was first explicitly stated by A. Cayley and W. R. Hamilton about in 1858, but the first general proof was published in 1878 by G. Frobenius, and numerous others have appeared since then, for example see [1,2]. From the structure theorem for finitely generated modules over a principal ideal domain it straightforwardly follows the Cayley-Hamilton theorem and the proposition that there exists a vector v in a finite dimensional linear space V such that and a linear transformation of V have the same minimal polynomial. In this note, we provide alternative proofs of these results by only utilizing the knowledge of linear algebras.
基金Supported by National Natural Science Foundation of China(No.12171142).
文摘For a nilpotent group G without π-torsion,and x,y ∈ G,if x^(n)=y^(n) for a T-number n,then x=y;if x^(m)y^(n)=y^(n)x^(m) for n-numbers m,n,then xy=yx.This is a wellknown result in group theory.In this paper,we prove two analogous theorems on matrices,which have independence significance.Specifically,let m be a given positive integer and A a complex square matrix satisfying that(i)all eigenvalues of A are nonnegative,and(i)rank A^(2)=rank A;then A has a unique m-th root X with rank X^(2)=rank X,all eigenvalues of X are nonnegative,and moreover there is a polynomial f(λ)with X=f(A).In addition,let A and B be complex n×n matrices with all eigenvalues nonnegative,and rank A^(2)=rank A,rank B^(2)=rank B;then(i)A=B when A^(r)=B^(r) for some positive integer r,and(i)AB=BA when A^(s)B^(t)=B^(t)A^(s) for two positive integers s and t.
基金Project supported by NSFC （11371124, 11301150） and the Natural Science Foundation of Henan Province of China （142300410134, 162300410066）.
文摘The automorphism group of G is determined, where G is a nonabelian p-group given by a central extension as 1→Zpm→G→Zp×…×Zp→1 such that its derived subgroup has order p.
基金Supported by the National Natural Science Foundation of China(Grant No.11771129,11971155,12071117).
文摘Let A be a completely decomposable homogeneous torsion-free abelian group of rank n(n≥2).Let m(n)=A×(a)be the split extension of A by an automorphismαwhich is a cyclic permutation of the direct components twisted by a rational integer m.Then Om(n)is an infinite soluble group.In this paper,the residual finiteness of Om(n)is investigated.
基金Supported by the National Natural Science Foundation of China(Grant No.11771129).
文摘Let Z>be a principal ideal domain(PID)and M be a module over D.We prove the following two dual results:(i)If M is finitely generated and rr,y are two elements in M such that M/Dx≌M/Dy,then there exists an auto morphism a of M such that a(x)=y.(ii)If M satisfies the minimal cond计ion on submodules and X,Y are two locally cyclic submodules of M such that M/X≌M/Y and X≌Y,then there exists an automorphism a of M such that α(X)=Y.
基金supported by the National Natural Science Foundation of China(Nos.11771129,11301150,11601121)the Natural Science Foundation of Henan Province of China(No.162300410066)
文摘Let G be a finite p-group with a cyclic Frattini subgroup. In this paper, the automorphism group of G is determined.
基金supported by National Natural Science Foundation of China(Grant No.12171142)。
文摘Let p be a prime and F be a finite field of characteristic p.Suppose that FG is the group algebra of the finite p-group G over the field F.Let V(FG)denote the group of normalized units in FG and let V_(*)(FG)denote the unitary subgroup of V(FG).If p is odd,then the order of V_(*)(FG)is|F|^((|G-1)/2).However,the case p=2 still is open.In this paper,the order of V*(FG)is computed when G is a nonabelian 2-group given by a central extension of the form 1→Z_(2^(n))×Z_(2^(m))→G→Z_(2)×…×Z_(2)→1 and G'≌Z_(2),n,m≥1.Furthermore,a conjecture is confirmed,i.e.,the order of V_(*)(FG)can be divisible by|F|^(1/2(|G|+|Ω1(G)|)-1),where Ω_(1)(G)={g∈G|g^(2)=1}.
基金Project supported by the NSFC （11301150, 11371124）, Natural Science Foundation of Henan Province of China （142300410134）, Program for Innovation Talents of Science and Technology of Henan University of Technology （11CXRC19）.
文摘Let G be a group. A subset X of G is said to be non-commuting if xy ≠ yx for any x, y ∈ X with x ≠ y. If {X}≥ IYI for any other non-commuting set Y in G, then X is said to be a maximal non-commuting set. In this paper, the bound for the cardinality of a maximal non-commuting set in a finite p-group G is determined, where G is a non-abelian p-group given by a central extension as1 → Zp→ G →Zp ×→ × Zp →1 and its derivedsubgroup has order p.
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.11801129,11771129)the Natural Science Foundation of Hebei Province(No.A2019402211)+3 种基金the Program for Young Top Talent of Higher Learning Institutions of Hebei(No.BJ2018025)the Foundation of Handan(No.1723208068-5)the Excellent Young and Middle-Aged Innovative Team Program of Hubei(No.T201601)the New Century High-Level Talents Foundation of Hubei.
文摘Let G be a group,and let a be a regular automorphism of order p2 of G,where p is a prime.If G is polycyclic-by-finite and the map φ:G→G defined by=g^φ,[g,a]is surjective,then G is soluble.If G is polycyclic,then CG(a^p)and G/[G,a^p]are both nilpotent-by-finite.
