In the book, using the method of exceptional characters using the Frobenius subgroup, a number of problems of the theory of finite groups are solved: group-theoretic characterization of simple Zassenhaus groups is obtained, the problem of A. I. Kostrikin on finite groups with large Sylow subgroups is solved, theorems are given on the kernel of an irreducible nature of a given degree, on groups with an complementable nilpotent subgroup, and on linear groups.