The Fitting length of a product of mutually permutable finite groups

Abstract

Let the finite soluble group $${G = G_{1}G_{2} \cdots G_{r}}$$ be the product of pairwise mutually permutable subgroups $${G_{1}, G_{2}, \ldots, G_{r}}$$ , let h(G) and $${\ell_{p}(G)}$$ be respectively the Fitting length and the p-length of G. The aim of this paper is to prove that $${h(G) \leq {\rm max} \{h(G_{i}) \mid i = 1, 2, \ldots, r\}+1}$$ and $${\ell_{p}(G) \leq {\rm max} \{\ell_{p}(G_{i}) \mid i = 1, 2, \ldots, r\}+1}$$ .