Abstract
Characterizations of some [Formula: see text]-groups are obtained: [Formula: see text]-groups of maximal class, metacyclic [Formula: see text]-groups. The critical subgroups for solvable groups are defined. For a number of known results new proofs are presented. For investigation of [Formula: see text]-automorphisms of [Formula: see text]-groups the minimal nonnilpotent groups are used essentially. Use of counting theorems allows us to simplify some proofs. A number of open problems is stated.
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