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中文摘要：
根据子群的性质来研究群的性质和结构是群论研究中的一个比较热门的课题。本文主要研究了λ可补充子群对有限群结构的影响，即一个群的子群的λ-可补充性可以确定这个群本身的p-幂零性和超可解性。通过考察群的极小子群或者4阶循环子群的λ-可补充性，本文给出了一个群是超可解群的充分必要条件：一个群G是超可解的当且仅当G有一个正规子群E使得G/E是超可解的，且对E的每个非循环的Sylow子群P，P的每个在G中无超可解补充的极小子群或者4阶循环子群H（如果P是一个非交换2-群，且HZSymboleB@ （G））在G中是λ-可补充的。在对群的p-幂零性给出了一个新刻画的基础上，应用极小阶反例法和数学归纳法证明了该充要条件。该结论推广并统一了部分已有文献的研究成果。
英文摘要：
It is an interesting topic in group theory to study the structure of groups whose subgroups possess some special proper ties. The present article is devoted to investigate the influence of minimal A-supplemented subgroups on the structure of groups, which shows that the p-nilpotency and the supersolubility of a group can be determined by its A-supplemented subgroups. This paper gives a necessary and sufficient condition for a group to be supersoluble in terms of the A-supplemented subgroups of orders prime or 4, that is a group G is supersoluble if and only if G has a normal subgroup E such that G/E is supersoluble and for every non-cyclic Sylow subgroup P of E, every cyclic subgroup H of P of orders prime or 4 （if is a non-abelian 2-group and HcZ∞（G）） without a supersoluble supplement in G is ）,-supplemented in G. In order to prove this result, a new characterization of p-nilpotency of groups is given and the minimal counterexamples are considered in this paper. As applications, some known results are generalized and uni- fied.
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Finite groups with the set of the number of subgroups of possible order containing exactly two eleme
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