As recounted in this paper, the idea of groups is one that has evolved from some very intuitive concepts. We can do binary operations like adding or multiplying two elements and also binary operations like taking the ...As recounted in this paper, the idea of groups is one that has evolved from some very intuitive concepts. We can do binary operations like adding or multiplying two elements and also binary operations like taking the square root of an element (in this case the result is not always in the set). In this paper, we aim to find the operations and actions of Lie groups on manifolds. These actions can be applied to the matrix group and Bi-invariant forms of Lie groups and to generalize the eigenvalues and eigenfunctions of differential operators on R<sup>n</sup>. A Lie group is a group as well as differentiable manifold, with the property that the group operations are compatible with the smooth structure on which group manipulations, product and inverse, are distinct. It plays an extremely important role in the theory of fiber bundles and also finds vast applications in physics. It represents the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. Here we did work flat out to represent the mathematical aspects of Lie groups on manifolds.展开更多
从极大子群、中心主因子和正规子群的G 主列的角度来讨论有限幂零群,获得了有限幂零群的若干新刻划.设M是有限群G的任一极大子群,H G.令 G=G/Φ(G),则G是幂零群当且仅当下列条件之一成立:(1)如H≤\M,则H∩M G且H/H∩M≤Z(G/H∩M);(2)如H...从极大子群、中心主因子和正规子群的G 主列的角度来讨论有限幂零群,获得了有限幂零群的若干新刻划.设M是有限群G的任一极大子群,H G.令 G=G/Φ(G),则G是幂零群当且仅当下列条件之一成立:(1)如H≤\M,则H∩M G且H/H∩M≤Z(G/H∩M);(2)如H≤\M,则M≤CG(H/H∩M);(3)如H≤\M,则H≤CG(M/H∩M);(4)如H≤\M,则M补于G的一个中心主因子;(5)F( G)有一个 G 主列,其中每个主因子都是 G中心的且CG(F( G))可解;(6)Soc( G)有一个 G 主列,其中每个主因子都是 G 中心的;(7)K∞(G)≤H,H/Φ(H)有一个 G 主列其中每个主因子都是 G 中心的;(8)HCG(H)≤Z∞(G).展开更多
文摘As recounted in this paper, the idea of groups is one that has evolved from some very intuitive concepts. We can do binary operations like adding or multiplying two elements and also binary operations like taking the square root of an element (in this case the result is not always in the set). In this paper, we aim to find the operations and actions of Lie groups on manifolds. These actions can be applied to the matrix group and Bi-invariant forms of Lie groups and to generalize the eigenvalues and eigenfunctions of differential operators on R<sup>n</sup>. A Lie group is a group as well as differentiable manifold, with the property that the group operations are compatible with the smooth structure on which group manipulations, product and inverse, are distinct. It plays an extremely important role in the theory of fiber bundles and also finds vast applications in physics. It represents the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. Here we did work flat out to represent the mathematical aspects of Lie groups on manifolds.
基金Supported by the Natural Science Foundation of China and the Natural Science Foundation of Guangxi Autonomous Region (No0249001 )the Graduate lnnovation Foundation of Guangxi Autonomous Region
文摘从极大子群、中心主因子和正规子群的G 主列的角度来讨论有限幂零群,获得了有限幂零群的若干新刻划.设M是有限群G的任一极大子群,H G.令 G=G/Φ(G),则G是幂零群当且仅当下列条件之一成立:(1)如H≤\M,则H∩M G且H/H∩M≤Z(G/H∩M);(2)如H≤\M,则M≤CG(H/H∩M);(3)如H≤\M,则H≤CG(M/H∩M);(4)如H≤\M,则M补于G的一个中心主因子;(5)F( G)有一个 G 主列,其中每个主因子都是 G中心的且CG(F( G))可解;(6)Soc( G)有一个 G 主列,其中每个主因子都是 G 中心的;(7)K∞(G)≤H,H/Φ(H)有一个 G 主列其中每个主因子都是 G 中心的;(8)HCG(H)≤Z∞(G).
