MOvan; @FMOpi]) extends previous developements by considering some special type of factorisations, the so-called *core-factorisations*. Only [@FMOprime] treats prime power indices without considering any additional restriction on the factors. On the other hand, it is to be said that in most cases the conditions on the indices are imposed only on some subsets of elements of the factors, namely prime power order elements, $p$-regular elements, zeros of irreducible characters, etc.

The notation and terminology are as follows. For an element $x$ in a group $G$, we call $i_G(x)$ the *index of $x$*, i.e $i_G(x)=|G:{{\operatorname}{C}_{G}(x)}|$. A *$p$-regular* element is an element whose order is not divisible by $p$, where $p$ will always be a prime number. If $n$ is a positive integer, $n_p$ denotes the highest power of $p$ dividing $n$. We represent by $\pi(G)$ the set of all prime divisors of $|G|$. The set of all Sylow $p$-subgroups of $G$ is ${{\operatorname}{Syl}_{p}\left(G\right)}$ and ${{\operatorname}{Hall}_{\pi}\left(G\right)}$ is the set of all Hall $\pi$-subgroups of $G$ for a set of primes $\pi$. A group such that $G={{\operatorname}{O}_{\pi}(G)}\times {{\operatorname}{O}_{\pi'}(G)}$ is said to be *$\pi$-decomposable*. If $H$ is a subgroup of $G$, we denote by $H^G$ the normal closure of $H$ in $G$. The remaining notation and terminology are standard within the theory of finite groups, and they mainly follow those of the book [@DH], apart from some terminology on simple groups which will be introduced later.

It is well known that if $p$ does not divide $i_G(x) $ for every $p$-regular element in a group $G$, then the Sylow $p$-subgroup is a direct factor of $G$ (see for instance [@CC Lemma 2]). This result was improved in [@LWW Theorem 5] by proving that the same conclusion remains true if the conditions on the indices are only imposed on $p$-regular elements of prime power order. In this paper, we deal with the corresponding result for factorised groups, but avoiding the consideration of any additional 