Determining Sets and Determining Numbers of Finite Groups

Abstract

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].