有限群广义表示的进一步研究

Further study on the generalized representation of finite groups

经典的群表示是通过研究与抽象群保持运算的具体群来获取原来群的结构的丰富信息,如通过研究与抽象群G保持同态的线性空间V的全体可逆线性变换组成的乘法群GL(V)及域F上全体可逆矩阵组成的乘法群GL(n,F),解决与群结构有关的许多问题.该文在已有定义及相关结论的基础上,继续研究群G到GL(V)或GL(n,F)的广义同态,即群的广义线性表示和广义矩阵表示,获得若干有用的群结构信息.

Classical group representation theory obtains rich information about the structure of the original group by studying the specific group that maintains the operation with the abstract group.For example, many problems related to the group structure are solved by studying the multiplication group GL(V)composed of all reversible linear transformations in the linear space that maintains homomorphism with the abstract group, and also the multiplication group GL(n,F)composed of all reversible matrices on the field F.Based on the existing definitions and relevant conclusions, this paper continues to study the generalized homomorphism of group G to GL(V)or GL(n,F),namely, the generalized linear representation and generalized matrix representation of groups, and obtains some useful group structure information.