The theory of infinite soluble groups.

*(English)*Zbl 1059.20001
Oxford Mathematical Monographs. Oxford Science Publications. Oxford: Clarendon Press (ISBN 0-19-850728-3/hbk). xvi, 342 p. (2004).

The theories of finite and of infinite soluble groups have gone through very different stages of development. Conditions trivially satisfied in finite groups lead to interesting subclasses of infinite groups, tools used in the infinite case are of different nature. Both theories have developed very much in the last seventy years. Finite soluble groups have been described extensively by Doerk and Hawkes. The authors have set out to give a survey on the results known in infinite soluble groups. They avoid subjects that were well covered in their opinion: ordered, profinite groups, group lattices, varieties, products of subgroups. Finitely generated groups is a repeated theme here, the subject is taken up again when the aspects considered in the meantime allow progress for this class. The aspects are nilpotent groups, linear groups, finite rank, finiteness conditions on Abelian groups.

Certain subclasses allow centrality statements (like having a bound on the central height of subgroups), they are considered, also algorithmic theories and cohomological methods. A section on finitely presented groups follows. The survey closes with a section on groups all subgroups of which are subnormal. The long development to the discovery that all of them are soluble is described here.

This book allows the reader to inform himself up to most recent research, after the guidance by the book he can take advantage of the bibliography which consists of 811 items dating from 1938 (Hirsch) to 2002 (Kochloukova, Smith).

Certain subclasses allow centrality statements (like having a bound on the central height of subgroups), they are considered, also algorithmic theories and cohomological methods. A section on finitely presented groups follows. The survey closes with a section on groups all subgroups of which are subnormal. The long development to the discovery that all of them are soluble is described here.

This book allows the reader to inform himself up to most recent research, after the guidance by the book he can take advantage of the bibliography which consists of 811 items dating from 1938 (Hirsch) to 2002 (Kochloukova, Smith).

Reviewer: H. Heineken (Würzburg)

##### MSC:

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

20F16 | Solvable groups, supersolvable groups |

20E07 | Subgroup theorems; subgroup growth |

20E15 | Chains and lattices of subgroups, subnormal subgroups |