Centraliser-Related Decompositions and Extension Types in the Lattice of Intermediate Subgroups of a Group Extension

In this paper, centraliser-related intermediate subgroups of group extensions G/F are explored in some depth. Four specific types of group extensions are distinguished according to the positioning of their centraliser-related intermediate groups. Examples of such group extensions are central group extensions G/F where the subgroup generated by F and the centraliser of F is equal to G, and inner extensions, where the double centraliser of F in G is equal to F. It is shown how each group extension can be decomposed in a canonical manner in terms of four intermediate extensions that are of these four specific types. In addition, some special cases are distinguished through which simpler forms of canonical decompositions are obtained. It is shown how the specific extension types and decompositions can be used to find out more about the structure of a group extension and how they have dual centraliser relations within a larger group A.