Finite group generated by Schur-trivial subgroups of relatively prime indices is Schur-trivial
Suppose is a finite group. Suppose are subgroups of such that the values of the indices , 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 is a Schur-trivial group. Then, the group is also a Schur-trivial group.