Determining sets and determining numbers of finite groups

الأصل · EN

Let G be a group. A subset D of G is a determining set of G, if every automorphism of G is uniquely determined by its action on D. The determining number of G, denoted by α(G), is the cardinality of a smallest determining set. A generating set of G is a subset such that every element of G can be expressed as the combination, under the group operation, of finitely many elements of the subset and their inverses. The cardinality of a smallest generating set of G, denoted by γ(G), is called the generating number of G. A group G is called a DEG-group if α(G)=γ(G). The main results of this article 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; A finite group is a normal subgroup of a DEG-group and there is an injective mapping from the set all finite groups to the set of finite DEG-groups; Nilpotent groups of order n which have the maximum determining number are classified; For any integer k≥ 2, there exists a group G such that α(G)=2 and γ(G)≥ k.