Science in China Series A: Mathematics Feb., 2009, Vol. 52, No. 2, 217–230 www.scichina.com math.scichina.com www.springerlink.com Factorizations of groups and related topics AMBERG Bernhard 1 † & KAZARIN Lev 2 1 Fachbereich 08, Universitaet Mainz, Germany 2 Department of Mathematics, Yaroslavl State University, 150000 Yaroslavl, Russia (email: amberg@mathematik.uni-mainz.de, kazarin@uniyar.ac.ru) Abstract This is a survey of some recent progresses in the theory of groups with factorizations. Some of the methods can be used to obtain information about finite groups in general, nilpotent algebras and nearrings. Keywords: MSC(2000): 20D40 1 Groups with factorizations A group G is factorized if G = AB is the product of two subgroups A and B, i.e., every element g of G can be expressed in the form g = ab for some a ∈ A, b ∈ B. The factorization is proper if A and B are proper subgroups of G. Factorizations of groups naturally arise from the Frattini-Lemma and also when G is a permutation group on a set Ω having a proper transitive subgroup. On the other hand, there are groups which have no proper factorizations with subgroups A and B, for instance the finite cyclic p-groups and the infinite quasicyclic Pr¨ ufer p-groups for a prime p. Groups with proper factorizations have been studied by many authors, and some well-known theorems are in fact the statements about factorized groups. The main problem concerning factorized groups is the following Problem 1.1. Let G = AB be a product of two subgroups A and B with given properties. What can be said about the structure of the factorized group G? In the following we are mainly concerned about some recent progresses on this general prob- lem in special situations. Some of these new results are only possible, since the Classification of Finite Simple Groups (CFSG) is available. The notation is standard and follows by [1, 2], p always denotes a prime. For further results about factorized groups we refer the reader to [3]; see also [4–10]. 2 Some classical results The following well-known p α -lemma by Burnside is perhaps the first important result about factorizations of groups. If G is a finite group and x ∈ G, we define the index i G (x)= |G : C G (x)|. Received July 20, 2008; accepted October 11, 2008 DOI: 10.1007/s11425-009-0024-8 † Corresponding author