On the Generalized Fitting Height and Nonsoluble Length of the Mutually Permutable Products of Finite Groups (2301.02199v1)

Abstract: 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 $h=\lambda_p(G)$ 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 $\max{h^*(A), h^*(B)}\leq h^*(G)\leq \max{h^*(A), h^*(B)}+1$ and $\max{\lambda_p(A), \lambda_p(B)}= \lambda_p(G)$. Also we introduced and studied the non-Frattini length.