关于有限群互换积的广义拟合高和不可解长
On the Generalized Fitting Height and Nonsoluble Length of the Mutually Permutable Products of Finite Groups
摘要：
有限群$G$的广义Fitting高度$h^*(G)$是使得$\mathrm{F}_h^* (G) = G$的最小数$h$，其中$\mathrm{F}_{(0)}^* (G) = 1$和$\mathrm{F}_{(i+1)}^*(G)$是广义Fitting子群$\mathrm{F}^*(G/\mathrm{F}^*_{(i)} (G))$的逆像。设$p$是素数，$1=G_0\leq G_1\leq\dots\leq G_{2h+1}=G$是最短正规数列，其中对于$i$奇数，因子$G_{i+1}/G_i$是$p$ -可解的（可能是平凡的），对于$i$偶数，因子$G_{i+1}/G_i$是非交换单群的（非空）直积。则$h=λ_p(G)$称为$G$基团的非$p$可溶性长度.证明了：若有限群$G$是子群$A$和$B$的互可置换积，则有$\max\{h^*(A), h^*(B)\}\leq h^*(G)\leq \max\{h^*(A), h^*(B)\}+1$和$\max\{λ_p(A), λ_p(B)\}= λ_p(G)$ .同时，我们还引入并研究了非Frattini长度。
英文摘要：
The generalized Fitting height $h^*(G)$ of a finite group $G$ is the least number $h$ such that $\mathrm{F}_h^* (G) = G$, where $\mathrm{F}_{(0)}^* (G) = 1$, and $\mathrm{F}_{(i+1)}^*(G)$ is the inverse image of the generalized Fitting subgroup $\mathrm{F}^*(G/\mathrm{F}^*_{(i)} (G))$. Let $p$ be a prime, $1=G_0\leq G_1\leq\dots\leq G_{2h+1}=G$ be the shortest normal series in which for $i$ odd the factor $G_{i+1}/G_i$ is $p$-soluble (possibly trivial), and for $i$ even the factor $G_{i+1}/G_i$ is a (non-empty) direct product of nonabelian simple groups. Then $h=λ_p(G)$ is called the non-$p$-soluble length of a group $G$. We proved that if a finite group $G$ is a mutually permutable product of of subgroups $A$ and $B$ then $\max\{h^*(A), h^*(B)\}\leq h^*(G)\leq \max\{h^*(A), h^*(B)\}+1$ and $\max\{λ_p(A), λ_p(B)\}= λ_p(G)$. Also we introduced and studied the non-Frattini length.
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