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Products of groups. (English) Zbl 0774.20001
Oxford Mathematical Monographs. Oxford: Clarendon Press. xii, 220 p. (1992).
This book collects the as yet known answers to the following general question: Let \(A,B\) be two groups and \(G=AB\) their product. What properties of \(AB\) can be derived from \(A\) and \(B\)? The first general result in this area was proved by Itô; he showed that \(AB\) is metabelian if \(A\) and \(B\) are abelian. This result is unparalleled in shortness of proof and absence of additional conditions. Another major milestone was the detection by Kegel and Wielandt that products of finite nilpotent groups are soluble.
The authors follow mostly the historical lines of development by organizing the book, after an introductory part, in chapters on products of nilpotent groups, of periodic groups, and of groups of finite rank. Further information on the structure is collected in the fifth chapter on splitting and conjugacy theorems. The material is completed by two chapters devoted to generalizations (one of them on triply factorized groups \(G=AB=BC=CA\)). The next steps one could think of are indicated by questions.
This book is a good source for anyone who wants to know about the situation in this area; the systematic arrangement eases the task for someone who looks for a particular result of Chernikov, Kazarin, Zaitzev and the authors – to mention only those contributors who are mentioned more than six times in the bibliography of around 170 entries.

MSC:
20-02 Research exposition (monographs, survey articles) pertaining to group theory
20E22 Extensions, wreath products, and other compositions of groups
20E34 General structure theorems for groups
20F19 Generalizations of solvable and nilpotent groups
20E07 Subgroup theorems; subgroup growth
20E15 Chains and lattices of subgroups, subnormal subgroups
20F16 Solvable groups, supersolvable groups
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