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,a(G),is the cardinality of a smallest determining set.A group G is called a DEG-group if α(G)equals(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(G)≥k.