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

Statement

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 $[G:H_{i}],1\leq i\leq n$, 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.

Related facts

• Cyclic implies Schur-trivial
• All Sylow subgroups are Schur-trivial implies Schur-trivial