# Finite group generated by Schur-trivial subgroups of relatively prime indices is Schur-trivial

Suppose $G$ is a finite group. Suppose $H_1,H_2,\dots,H_n$ are subgroups of $G$ such that the values of the indices $[G:H_i], 1 \le i \le n$, have gcd 1 (note that we do not assume that any two of them are relatively prime -- we only assume that there is no prime factor common to all the indices). Suppose further, that each $H_i$ is a Schur-trivial group. Then, the group $G$ is also a Schur-trivial group.