论文标题
有限群体承认满足额外多项式身份的企业自动形态
Finite groups admitting a coprime automorphism satisfying an additional polynomial identity
论文作者
论文摘要
众所周知，具有自动形态$φ$的有限群体的可溶性自由基为$（|φ|，| c_g（φ）|）$ - 有限的拟合高度和索引。我们扩展了这个经典结果，如下所示。令$ f（x）= a_0 + a_1 \ cdot x + \ cdots + a _d \ cdot x^d \ in \ mathbb {z} [z} [x] $是一个原始的多项式，让$ g $为有限的群体，是一个有限型组，带有自动化订单的$ g^$ g^cdot cdot cdot cd {a_0} a_0} a_0} {a_0} \cdotsφ^d（g）^{a_d} = 1 $，对于g $中的所有$ g \。然后，$ g $的可溶性根部具有$（d，| c_g（φ）|）$ - 绑定拟合高度和索引。界限是明确的，对于$ d $的少量值特别有利。
It is known that a finite group with an automorphism $φ$ of coprime order has a soluble radical of $(|φ|,|C_G(φ)|)$-bounded Fitting height and index. We extend this classic result as follows. Let $f(x) = a_0 + a_1 \cdot x + \cdots + a_d \cdot x^d \in \mathbb{Z}[x]$ be a primitive polynomial and let $G$ be a finite group with an automorphism $φ$ of coprime order satisfying $ g^{a_0} \cdot φ(g)^{a_1} \cdots φ^d(g)^{a_d} = 1 $, for all $g \in G$. Then the soluble radical of $G$ has $(d,|C_G(φ)|)$-boundex Fitting height and index. The bounds are made explicit and are particularly good for small values of the degree $d$.
