The Research On Nilpotency Of Finite Groups

One of the important problems in the theory of finite groups is to research the effect of nilpotency on the finite groups. In 1962, Ito introduced: if G is a finite group and the order of G is an odd, every minimal groups of G is contained in Z(G), then G is nilpotent. Many experts of group theory are in favor in the nilpotency of finite groups and have got lots of results (see[1], Charpterâ…£). But they had less results about the cover and avoidance properties and semi cover and avoidance properties and the nilpotency of the finite groups. In this paper, we study about the cover and avoidance properties and semi cover and avoidance properties and the nilpotency of the finite groups.In 1962, Gaschutz introduced a certain conjugacy class of subgroups which have the cover and avoidance properties of a finite solvable group (see[2]). Thereafter many authors studied this property, for example, Gillam and Tomkinson(see [3][4]). In these papers, the main aim was to find some kind of subgroups of a finite soluble group G having the cover-avoidance properties. However, the questions arises whether we can obtain structural insight into a finite group when some of its subgroups have the cover-avoidance properties. In 1993, Ezquerro gave some characterization for a finite group G to be p-supersolvable and supersolvable based on the assumption that all maximal subgroups of G have the cover and avoidance properties. Thereafter, Xiuyun Guo, Yun Fan, Xianying He pushed further this approach and got some results about the cover and avoidance properties and semi cover and avoidance properties and the solvability and supersolvability and p-nilpotency of the finite groups. In this paper, we will push further this approach and get some results about the cover and avoidance properties and semi cover and avoidance properties and the nilpotency of the finite groups.