On the Generalized Fitting Height and Nonsoluble Length of the Mutually Permutable Products of Finite Groups

Abstract

The generalized Fitting height $h^{\ast }(G)$ of a finite group $G$ is the least number $h$ such that ${\mathrm{F}}_h^{\ast }(G)=G$, where ${\mathrm{F}}_{(0)}^{\ast }(G)=1$, and ${\mathrm{F}}_{(i+1)}^{\ast }(G)$ is the inverse image of the generalized Fitting subgroup ${\mathrm{F}}^{\ast }(G/{\mathrm{F}}_{(i)}^{\ast }(G))$. Let $p$ be a prime, $1=G_0\le G_1\le \cdots \le G_{2h+1}=G$ 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 $h={\lambda }_p(G)$ 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 $\max \{h^{\ast }(A),h^{\ast }(B)\}\le h^{\ast }(G)\le \max \{h^{\ast }(A),h^{\ast }(B)\}+1$ and $\max \{{\lambda }_p(A),{\lambda }_p(B)\}={\lambda }_p(G)$. Also we introduced and studied the non-Frattini length.

arXiv admin note: substantial text overlap with arXiv:2103.13354