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

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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 \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.

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