希尔伯特定理90的群论版本
A group theoretical version of Hilbert's theorem 90
摘要：
证明了对于群$G$，$G/N$循环的正规子群$N$，映射$N^{\mathrm{ab}}\to G^{\mathrm{ab}}$的核满足经典的Hilbert 90性质（参见. Thm. A）。因此，如果$G$是无扭生成的，$|G:N|<\infty$，并且所有阿贝尔群$H^{\mathrm{ab}}$，$N\subseteq H\subseteq G$都是无扭的，那么$N^{\mathrm{ab}}$必须是$G/N$的伪置换模（参见。Thm. B）。从定理A还推导出传递核和余核的阶之间的非平凡关系，其确定希尔伯特-铃木乘子（cf. Thm. C）。转换到数论的背景下，人们得到希尔伯特定理94的强形式。在$G$是随机生成的并且$N$在$G$中具有素数索引$p$的情况下，存在“广义Schreier公式”，其涉及$G$和$N$的无挠秩以及传递核和协核的阶的比率（参见图10）。Thm. D）。
英文摘要：
It is shown that for a normal subgroup $N$ of a group $G$, $G/N$ cyclic, the kernel of the map $N^{\mathrm{ab}}\to G^{\mathrm{ab}}$ satisfies the classical Hilbert 90 property (cf. Thm. A). As a consequence, if $G$ is finitely generated, $|G:N|<\infty$, and all abelian groups $H^{\mathrm{ab}}$, $N\subseteq H\subseteq G$, are torsion free, then $N^{\mathrm{ab}}$ must be a pseudo permutation module for $G/N$ (cf. Thm. B). From Theorem A one also deduces a non-trivial relation between the order of the transfer kernel and co-kernel which determines the Hilbert-Suzuki multiplier (cf. Thm. C). Translated into a number theoretic context one obtains a strong form of Hilbert's theorem 94. In case that $G$ is finitely generated and $N$ has prime index $p$ in $G$ there holds a "generalized Schreier formula" involving the torsion free ranks of $G$ and $N$ and the ratio of the order of the transfer kernel and co-kernel (cf. Thm. D).
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