Products of Finite Groups

Finite groups that can be decomposed as products of (pairwise permutable) subgroups have a rich and deep structure. In hindsight, many classical results can be seem to point to this theme: From Burnside-Wielandt’s theorem that a finite group G is nilpotent if and only if G is the product of its Sylow subgroups to P. Hall’s theorem that a finite group is solvable whenever it is a product of (pairwise permutable) Sylow subgroups, or more recently to Wielandt (1958) and Kegel (1962) theorem that every finite group that is a product of pairwise permutable nilpotent subgroups is solvable.

The monograph under review has as its main goal to study the structure of finite groups which can be decomposed as products of pairwise permutable subgroups. Thus, in addition to recovering many of the classical results, it gives a unified account of the main techniques and results, collecting in one place many of the more recent developments, with special focus on products of finite nilpotent groups and products of mutually and totally permutable subgroups. The monograph will be of interest to any researcher or graduate student working on finite group theory in the post-classification era.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx.