论文标题
有限组的拟合高度承认满足额外多项式身份的无定点自动形态
Fitting height of finite groups admitting a fixed-point-free automorphism satisfying an additional polynomial identity
论文作者
论文摘要
令$ f（x）$为具有整数系数的非零多项式。据说，如果每个特征性的基本Abelian $ S $ of $ g $的线性转换$ f（x）$，则据称是$ g $ $ g $的$ g $ $φ$。我们证明，如果有限的（可溶）$ g $承认满足小学Abelian身份$ f（x）$的无定点自动形态$φ我们还证明，如果$ f（x）$是任何非零的多项式，而$ g $是$σ'$ - 用于有限的Primes $σ=σ（f（x）$仅根据$ f（x）$，则$ g $的拟合高度的数量$ \ opersornArname $ \ red（x）（f（f（x）$）的拟合高度$ \ opertornareame $ rest crest vicomptional v（f（x）$）（f（x））（x）（x）（x） $ f（x）$。对于$ \langleφ\ rangle $，$ \ operatoTorname {deg}（f（x）$或$ \ operatateOnnAme {irrOnnAme {irrAneAme}（f（x）$相比，与$ $ $或φ相比，
Let $f(x)$ be a non-zero polynomial with integer coefficients. An automorphism $φ$ of a group $G$ is said to satisfy the elementary abelian identity $f(x)$ if the linear transformation induced by $φ$ on every characteristic elementary abelian section $S$ of $G$ is annihilated by $f(x)$. We prove that if a finite (soluble) group $G$ admits a fixed-point-free automorphism $φ$ satisfying an elementary abelian identity $f(x)$, where $f(x)$ is a primitive polynomial, then the Fitting height of $G$ is bounded in terms of $\operatorname{deg}(f(x))$. We also prove that if $f(x)$ is any non-zero polynomial and $G$ is a $σ'$-group for a finite set of primes $σ=σ(f(x))$ depending only on $f(x)$, then the Fitting height of $G$ is bounded in terms of the number $\operatorname{irr}(f(x))$ of irreducible factors in the decomposition of $f(x)$. These bounds for the Fitting height are stronger than the well-known bounds in terms of the composition length $α(|φ|)$ of $\langleφ\rangle$ when $\operatorname{deg} (f(x))$ or $\operatorname{irr}(f(x))$ is small in comparison with $α(|φ|)$.
