On dual pronormal subgroups and Fitting classes

Summary

Introduction

All groups considered in this note are finite.

The analysis of the possible embedding properties of the subgroups in a group is a first way of entering into its structure. Normality and subnormality are the most elementary ones. From the study of conjugacy classes of subgroups the property of pronormality arises. A subgroup H of a group G is said to be pronormal in G if, for every element g of G, H and H^g are conjugated in their join 〈H,H^g〉. In [17] it was proved that in fact this condition is equivalent to their conjugacy in 〈H,H^g〉^N, the smallest normal subgroup of 〈H,H^g〉 with nilpotent factor group. This fact motivated the concept of dual pronormality. A subgroup H of a group G is said to be dual pronormal in G if, for every element g of G, F(〈H,H^g〉), the Fitting subgroup of 〈H,H^g〉, is contained in H. This property emerges both as a weaker condition than normality and as a dual concept to pronormality. Its influence on the structure of the groups was initially studied in a series of papers ([4], [5], [6]). Dual pronormal subgroups are close to N-injectors, for the class N of nilpotent groups, and, in this study, groups containing several relevant classes of dual pronormal subgroups were also taken into consideration. This development shows how far dual pronormality is from normality and subnormality and how dual pronormality can provide additional information.