标题： 小群的分离诺特数 显示英文标题

标题： The separating Noether number of small groups

摘要： 分离诺特数是有限群的最小正整数$d$，使得对于该群的任何有限维复线性表示，任何两个不同的轨道都可以由次数不超过$d$的多项式不变量区分。 在本文中，确定了所有阶小于$32$的群、所有具有指数至多$2$的循环子群的有限群以及二面体群与$2$元素群的直积的精确值。 结果特别表明，与普通诺特数不同，非交换有限群的分离诺特数可能等于该群的真直因子的分离诺特数。 大部分结果适用于包含一个乘法阶等于群大小的元素的一般（可能有限）基域的情况。 显示英文摘要

摘要： The separating Noether number of a finite group is the minimal positive integer $d$ such that for any finite dimensional complex linear representation of the group, any two dictinct orbits can be distinguished by a polynomial invariant of degree at most $d$. In the present paper its exact value is determined for all groups of order less than $32$, for all finite groups with a cyclic subgroup of index at most $2$, and for the direct product of a dihedral group with the $2$-element group. The results show in particular that unlike the ordinary Noether number, the separating Noether number of a non-abelian finite group may well be equal to the separating Noether number of a proper direct factor of the group. Most of the results are proved for the case of a general (possibly finite) base field containing an element whose multiplicative order equals the size of the group.