On the solvability and factorization of some $\pi$-solvable irreducible linear groups of primary degree. Part I

The article begins a series of papers where for a set $\pi$ of odd primes $\pi$-solvable finite irreducible complex linear groups of degree $2|H|+1$ whose Hall $\pi$-subgroups are $TI$-subgroups and are not normal in groups. The goal of this series is to prove the solvability and determine the factorization of such groups. Proof of the theorem started. Preliminary results are obtained and some properties of minimal counterexample to the theorem are established.