Let p be an odd prime,and let k be a nonzero nature number.Suppose that nonabelian group G is a central extension as follows1→G’→G→Z_(pK)×…×Z_(pK),where G’≌Zpk,andζG/G’is a,direct factor of G/G’.Th...Let p be an odd prime,and let k be a nonzero nature number.Suppose that nonabelian group G is a central extension as follows1→G’→G→Z_(pK)×…×Z_(pK),where G’≌Zpk,andζG/G’is a,direct factor of G/G’.Then G is a central product of an extraspecial pkgroup E andζG.Let|E|=p(2n+1)k and|ζG|=p(m+1)k.Suppose that the exponents of E andζG are pk+l and pk+r,respectively,where 0≤l,r≤k.Let AutG’G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G’,let AutG/ζG,ζG G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the centerζG and let AutG/ζG,ζG/G’G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially onζG/G’.Then(ⅰ)The group extension 1→Aut G’→Aut G→Aut G’→1 is split.(ⅱ)AutG’G/AutG/ζG,ζG G≌G1×G2,where Sp(2n-2,Zpk)■H≤G1≤Sp(2n,Zpk),H is an extraspecial pk-group of order p(2n-1)k and(GL(m-1,Zpk)■Zpk(m-1)■Zpk(m)≤G2≤GL(m,Zpk)■Zpk(m).In particular,G1=Sp(2n-2,Zpk)■H if and only if l=k and r=0;G1=Sp(2n,Zpx)if and only if l≤r;G2=(GL(m-1,Zpk)■Zpk(m-1)■Zpk(m)if and only if r=k;G2=GL(m,Zpk)■Zpk((m))if and only if r=0.(ⅲ)AutG’G/Aut G/ζG,ζG/G’G≌G1×G3,where G1 is defined in(ⅱ);GL(ml,Zpk)■Zpk(m-1)≤G3≤GL(n,Zpk).In particular,G3=GL(m-1,Zpk)■Zpk(m-1)if and only if r=k;G3=GL(m,Zpk)if and only if r=0.(ⅳ)AntG/ζG,ζG/G’G≌AutG/ζG,ζG/G’G■Zpk(m),If m=0,then AntG/ζG,ζG/G’G=Inn G≌Zpk(2n);If m>0,then AntG/ζG,ζG/G’G≌Zpk(2nm)×Zpk-r(2n),and AutG/ζG,ζG G/Inn G≌Zpk((2n(m-1))×Zpk-r(2n).展开更多
We discuss two different procedures to study the half Riordan arrays and their inverses.One of the procedures shows that every Riordan array is the half Riordan array of a unique Riordan array.It is well known that ev...We discuss two different procedures to study the half Riordan arrays and their inverses.One of the procedures shows that every Riordan array is the half Riordan array of a unique Riordan array.It is well known that every Riordan array has its half Riordan array.Therefore,this paper answers the converse question:Is every Riordan array the half Riordan array of some Riordan arrays?In addition,this paper shows that the vertical recurrence relation of the column entries of the half Riordan array is equivalent to the horizontal recurrence relation of the original Riordan array’s row entries.展开更多
The intersection of particular subgroups is a kind of interesting substructure in group theory. Let G be a finite group and D(G) be the intersection of the normalizers of the derived subgroups of all the subgroups of ...The intersection of particular subgroups is a kind of interesting substructure in group theory. Let G be a finite group and D(G) be the intersection of the normalizers of the derived subgroups of all the subgroups of G. A group G is called a D-group if G = D(G). In this paper, we determine the nilpotency class of the nilpotent residual G^(N) and investigate the structure of D(G) by a new concept called the IO-D-group. A non-D-group G is called an IO-D-group(inner-outer-D-group) if all of its proper subgroups and proper quotient groups are D-groups. The structure of IO-D-groups are described in detail in this paper. As an application of the classification of IO-D-groups, we prove that G is a D-group if and only if any subgroup of G generated by3 elements is a D-group.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
基金Supported by NSFC(Grant Nos.11601121,11771129)Natural Science Foundation of He’nan Province of China(Grant No.162300410066)。
文摘Let p be an odd prime,and let k be a nonzero nature number.Suppose that nonabelian group G is a central extension as follows1→G’→G→Z_(pK)×…×Z_(pK),where G’≌Zpk,andζG/G’is a,direct factor of G/G’.Then G is a central product of an extraspecial pkgroup E andζG.Let|E|=p(2n+1)k and|ζG|=p(m+1)k.Suppose that the exponents of E andζG are pk+l and pk+r,respectively,where 0≤l,r≤k.Let AutG’G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G’,let AutG/ζG,ζG G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the centerζG and let AutG/ζG,ζG/G’G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially onζG/G’.Then(ⅰ)The group extension 1→Aut G’→Aut G→Aut G’→1 is split.(ⅱ)AutG’G/AutG/ζG,ζG G≌G1×G2,where Sp(2n-2,Zpk)■H≤G1≤Sp(2n,Zpk),H is an extraspecial pk-group of order p(2n-1)k and(GL(m-1,Zpk)■Zpk(m-1)■Zpk(m)≤G2≤GL(m,Zpk)■Zpk(m).In particular,G1=Sp(2n-2,Zpk)■H if and only if l=k and r=0;G1=Sp(2n,Zpx)if and only if l≤r;G2=(GL(m-1,Zpk)■Zpk(m-1)■Zpk(m)if and only if r=k;G2=GL(m,Zpk)■Zpk((m))if and only if r=0.(ⅲ)AutG’G/Aut G/ζG,ζG/G’G≌G1×G3,where G1 is defined in(ⅱ);GL(ml,Zpk)■Zpk(m-1)≤G3≤GL(n,Zpk).In particular,G3=GL(m-1,Zpk)■Zpk(m-1)if and only if r=k;G3=GL(m,Zpk)if and only if r=0.(ⅳ)AntG/ζG,ζG/G’G≌AutG/ζG,ζG/G’G■Zpk(m),If m=0,then AntG/ζG,ζG/G’G=Inn G≌Zpk(2n);If m>0,then AntG/ζG,ζG/G’G≌Zpk(2nm)×Zpk-r(2n),and AutG/ζG,ζG G/Inn G≌Zpk((2n(m-1))×Zpk-r(2n).
文摘We discuss two different procedures to study the half Riordan arrays and their inverses.One of the procedures shows that every Riordan array is the half Riordan array of a unique Riordan array.It is well known that every Riordan array has its half Riordan array.Therefore,this paper answers the converse question:Is every Riordan array the half Riordan array of some Riordan arrays?In addition,this paper shows that the vertical recurrence relation of the column entries of the half Riordan array is equivalent to the horizontal recurrence relation of the original Riordan array’s row entries.
基金supported by National Natural Science Foundation of China (Grant Nos. 11631001 and 12071181)。
文摘The intersection of particular subgroups is a kind of interesting substructure in group theory. Let G be a finite group and D(G) be the intersection of the normalizers of the derived subgroups of all the subgroups of G. A group G is called a D-group if G = D(G). In this paper, we determine the nilpotency class of the nilpotent residual G^(N) and investigate the structure of D(G) by a new concept called the IO-D-group. A non-D-group G is called an IO-D-group(inner-outer-D-group) if all of its proper subgroups and proper quotient groups are D-groups. The structure of IO-D-groups are described in detail in this paper. As an application of the classification of IO-D-groups, we prove that G is a D-group if and only if any subgroup of G generated by3 elements is a D-group.
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.
