Theoretical Researches about u-Maximal Subgroups and Its Applications in Charactering IntuG

Let G be a finite group and u be the class of all finite supersoluble groups. A supersoluble subgroup U of G is called u-maximal in G if for any supersoluble subgroup V of G containing U, V=U. Moreover, IntuG is the intersection of all u-maximal subgroups of G. This paper obtains some new criteria on IntuG, by assuming that some subgroups of G are either Φ-I-supplemented or Φ-I-embedded in G. Here, a subgroup H of G is called Φ-I-supplemented in G if there exists a subnormal subgroup T of G such that G=HT and H∩THG/HG≤ΦH/HGIntuG and Φ-I-embedded in G if there exists a S-quasinormal subgroup T of G such that HT is S-quasinormal in G and H∩THG/HG≤ΦH/HGIntuG.

Ensuring a finite group is supersoluble

A special case of the main result is the following. Let G be a finite, non-supersoluble group in which from arbitrary subsets X, Y of cardinality n we can always find x ∈ X and y ∈ Y generating a supersoluble subgroup. Then the order of G is bounded by a function of n. This result is a finite version of one line of development of B.H. Neumann's well-known and much generalised result of 1976 on infinite groups.