On $\pi$-solvable irreducible linear groups with Hall $TI$-subgroup of odd order. I

The article begins a series of papers where for a set $\pi$ of odd primes $\pi$-solvable finite irreducible complex linear groups whose Hall $\pi$-subgroups are $TI$-subgroups and the degree of the group is small with respect to the order of such subgroup, are investigated. The goal of this series is to determine the possible values of the degree $n$ if a Hall $\pi$-subgroup $H$ is not normal and $n<2|H|$. The proof of a theorem that yields the complete list of these values is started. Some preliminary results are obtained and a number of properties of a minimal counterexample to the theorem is established.