论文标题
有限场上的代数组：亚组与同基因之间的连接
Algebraic Groups over Finite Fields: Connections Between Subgroups and Isogenies
论文作者
论文摘要
令G为在有限字段F_Q上定义的线性代数组。我们介绍了G的同基因与有限点G（F_Q^n）之间的几个连接。我们表明，从g'到g的同一基因在f_q上的g'g（f_q^n）在无限的许多n中产生了固定索引的子组。相反，我们表明，如果G是还原性，则无限许多n的固定指数K子组的存在意味着存在k级的同一基础。特别是，我们表明每个无限的亚组序列都受到有限数量的同基因的控制。该结果适用于GLM，SLM，SOM，SUM，SP2M的经典组，如果K是特征的主要，则可以扩展到非还原组。作为一种特殊情况，我们看到，如果G（g（f_q^n）与无穷大的g（f_q^n）的适当亚组的最小索引相连。通过改变特征p，研究了序列G（F_P）的类似结果。
Let G be a linear algebraic group defined over a finite field F_q. We present several connections between the isogenies of G and the finite groups of rational points G(F_q^n). We show that an isogeny from G' to G over F_q gives rise to a subgroup of fixed index in G(F_q^n) for infinitely many n. Conversely, we show that if G is reductive the existence of a subgroup of fixed index k for infinitely many n implies the existence of an isogeny of order k. In particular, we show that every infinite sequence of subgroups is controlled by a finite number of isogenies. This result applies to classical groups GLm, SLm, SOm, SUm, Sp2m and can be extended to non-reductive groups if k is prime to the characteristic. As a special case, we see that if G is simply connected the minimal indexes of proper subgroups of G(F_q^n) diverge to infinity. Similar results are investigated regarding the sequence G(F_p) by varying the characteristic p.
