Determining Sets and Determining Numbers of Finite Groups

A subset Formula: see text of a group Formula: see text is a determining set of Formula: see text if every automorphism of Formula: see text is uniquely determined by its action on Formula: see text, and the determining number of Formula: see text, Formula: see text, is the cardinality of a smallest determining set. A group Formula: see text is called a DEG-group if Formula: see text equals Formula: see text, the generating number of Formula: see text. Our main results are as follows. Finite groups with determining number 0 or 1 are classified; finite simple groups and finite nilpotent groups are proved to be DEG-groups; for a given finite group Formula: see text, there is a DEG-group Formula: see text such that Formula: see text is isomorphic to a normal subgroup of Formula: see text and there is an injective mapping from the set of all finite groups to the set of finite DEG-groups; for any integer Formula: see text, there exists a group Formula: see text such that Formula: see text and Formula: see text.