Determining Sets and Determining Numbers of Finite Groups.

A subset D of a group G is a determining set of G if every automorphism of G is uniquely determined by its action on D , and the determining number of G , α (G) , is the cardinality of a smallest determining set. A group G is called a DEG-group if α (G) equals γ (G) , the generating number of G. 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 H , there is a DEG-group G such that H is isomorphic to a normal subgroup of G and there is an injective mapping from the set of all finite groups to the set of finite DEG-groups; for any integer k ≥ 2 , there exists a group G such that α (G) = 2 and γ (G) ≥ k. [ABSTRACT FROM AUTHOR]