Abstract: Let G be a multiplicative group, B be a subset of G and h>=2 be an integer. Denote by B^h the set of all products of h elements of B . The set B is a basis of order h for G if B^h=G . In this paper the following result is proved: if G is a finite group of order n, then there exists a subset B of Gsuch that B^h=G and ｜B｜<= h(1- 1／ h )^1／h(n ·log n)^1／h+o(n^1／h) , where｜B｜is the cardinality of the subset B .

Key words: basis, finite group, cardinality

摘要： 设G为乘法群，B为G的子集，h为不小于2的整数，B^h为B中h各元素之积所成的集合。若B^h=G，则称B为G的阶为h的基。本文证明了如下结论：若G为n阶群，则存在G的子集B，使得B^h=G，|B|≤h(1-(1/h))^1/h(nlogn)^1/h+o(n)^1/h),其中|B|为子集B的基数。

关键词: 基, 有限群, 基数