调研了美国Class Central网站发布的“2023年250门全球最受欢迎在线课程榜”19门英语写作类MOOC课程。How to Write an Essay被列为250门全球最受欢迎MOOC课程之一,源于其优质的教学设计:世界顶尖大学及其丰富的在线课程经验、优秀师资...调研了美国Class Central网站发布的“2023年250门全球最受欢迎在线课程榜”19门英语写作类MOOC课程。How to Write an Essay被列为250门全球最受欢迎MOOC课程之一,源于其优质的教学设计:世界顶尖大学及其丰富的在线课程经验、优秀师资,教学目标注重培养学习者技能、终身学习理念以及抛锚式教学模式,这对于在线课程建设和教学具有借鉴意义。展开更多
Let G be a finite group, H ≤ G and R be a commutative ring with an identity 1R. Let CRG（H）={α ∈ RG｜αh = hα for all h ∈ H）, which is called the centralizer subalgebra of H in RG. Obviously, if H=G then CRG（H...Let G be a finite group, H ≤ G and R be a commutative ring with an identity 1R. Let CRG（H）={α ∈ RG｜αh = hα for all h ∈ H）, which is called the centralizer subalgebra of H in RG. Obviously, if H=G then CRG（H） is just the central subalgebra Z（RG） of RG. In this note, we show that the set of all H- conjugacy class sums of G forms an R-basis of CRG（H）. Furthermore, let N be a normal subgroup of G and γthe natural epimorphism from G to G to G/N. Then γ induces an epimorphism from RG to RG, also denoted by % We also show that if R is a field of characteristic zero, then γ induces an epimorphism from CRG（H） to CRG（H）, that is, 7（CRG（H）） = CRG（H）.展开更多
Let F be a number field and p be a prime. In the successive approximation theorem, we prove that, for each integer n ≥ 1, finitely many candidates for the Galois group of the nth stage of the p-class tower over F are...Let F be a number field and p be a prime. In the successive approximation theorem, we prove that, for each integer n ≥ 1, finitely many candidates for the Galois group of the nth stage of the p-class tower over F are determined by abelian type invariants of p-class groups C1pE of unramified extensions E/F with degree [E : F] = pn-1. Illustrated by the most extensive numerical results available currently, the transfer kernels (TE, F) of the p-class extensions TE, F : C1pF → C1pE from F to unramified cyclic degree-p extensions E/F are shown to be capable of narrowing down the number of contestants significantly. By determining the isomorphism type of the maximal subgroups S G of all 3-groups G with coclass cc(G) = 1, and establishing a general theorem on the connection between the p-class towers of a number field F and of an unramified abelian p-extension E/F, we are able to provide a theoretical proof of the realization of certain 3-groups S with maximal class by 3-tower groups of dihedral fields E with degree 6, which could not be realized up to now.展开更多
文摘调研了美国Class Central网站发布的“2023年250门全球最受欢迎在线课程榜”19门英语写作类MOOC课程。How to Write an Essay被列为250门全球最受欢迎MOOC课程之一,源于其优质的教学设计:世界顶尖大学及其丰富的在线课程经验、优秀师资,教学目标注重培养学习者技能、终身学习理念以及抛锚式教学模式,这对于在线课程建设和教学具有借鉴意义。
基金The NSF(11071155) of Chinathe NSF(2008A03) of Shandong Province
文摘Let G be a finite group, H ≤ G and R be a commutative ring with an identity 1R. Let CRG（H）={α ∈ RG｜αh = hα for all h ∈ H）, which is called the centralizer subalgebra of H in RG. Obviously, if H=G then CRG（H） is just the central subalgebra Z（RG） of RG. In this note, we show that the set of all H- conjugacy class sums of G forms an R-basis of CRG（H）. Furthermore, let N be a normal subgroup of G and γthe natural epimorphism from G to G to G/N. Then γ induces an epimorphism from RG to RG, also denoted by % We also show that if R is a field of characteristic zero, then γ induces an epimorphism from CRG（H） to CRG（H）, that is, 7（CRG（H）） = CRG（H）.
文摘Let F be a number field and p be a prime. In the successive approximation theorem, we prove that, for each integer n ≥ 1, finitely many candidates for the Galois group of the nth stage of the p-class tower over F are determined by abelian type invariants of p-class groups C1pE of unramified extensions E/F with degree [E : F] = pn-1. Illustrated by the most extensive numerical results available currently, the transfer kernels (TE, F) of the p-class extensions TE, F : C1pF → C1pE from F to unramified cyclic degree-p extensions E/F are shown to be capable of narrowing down the number of contestants significantly. By determining the isomorphism type of the maximal subgroups S G of all 3-groups G with coclass cc(G) = 1, and establishing a general theorem on the connection between the p-class towers of a number field F and of an unramified abelian p-extension E/F, we are able to provide a theoretical proof of the realization of certain 3-groups S with maximal class by 3-tower groups of dihedral fields E with degree 6, which could not be realized up to now.
