On Isaacs' problem

Let $G$ be a $\pi$-soluble irreducible complex linear group of degree $n$ such that a Hall $\pi$-subgroup $H$ of it has odd order, is a $\mathrm{TI}$-subgroup, and is not normal in $G$. In this paper it is established that $n$ is divisible by $|H|$ or by a power $f>1$ of some prime number such that $f\equiv \pm 1\ (\operatorname{mod}|H|)$.
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