Some results on finite trifactorized groups

Abstract

Let G be a finite and soluble group factorized by three subgroups A, B and C, that is $G= A B C=\{ abc \mid a \in A, b \in B, c \in C \}$, with $A B=B A$, $B C=C B$ and $C A=A C$. Let h(G) denote the Fitting length of G. In this paper we prove that if AB and C are nilpotent, then $h(G) \le h(B C) \cdot h(C A)$. We also prove that if $(\vert A \vert , \vert B \vert )=1$ and $\max \{ h(A B), h(B C), h(C A) \} = h$, then $h(G) \le h(h+1)$.