Products of groups with three generators

Question

In this question it mentions how Jesse Douglas used a Zappa-Szep product to classify some finite groups with two generators, and others did the same thing with infinite cyclic groups. I've been playing with a similar notion to the Zappa-Szep product but that multiplies three groups (beyond the trivial case where it can be reduced to two pairwise products of groups). Is anyone aware of existing literature on trying to classify three-generator groups, or some subtypes of them? All I was able to find were http://arxiv.org/pdf/math/9809174 and http://projecteuclid.org/euclid.mmj/1030132536, which look at Large Type and Finite Type Artin groups, respectively.

(If this question is too broad for this site, let me know.)

Answer 1

It seems that in the general one cannot say much about groups decomposable into a product of $>2$ subgroups. People have studied various special cases, e.g., when the factors are pairwise permutable. For example:

• J. Mennicke, Einige endliche Gruppen mit drei Erzeugenden und drei Relationen, Arch. Math. 10 (1959), 409--418.
• N.S. Chernikov, Groups that decompose into a product of permutable subgroups, Naukova Dumka, Kiev, 1987 (in Russian).
• A. Ballester-Bolinches et al., Products of Finite Groups, De Gruyter, 2010; DOI:10.1515/9783110220612 MR:2012d:20044