Stable groups. Transl. from the French by Moses Gabriel Klein (Q2723260)

This is a book that I like. When the French original ``Groupes stables'' (Zbl 0633.03019) appeared in 1987, it contained almost everything known about stable groups, from elementary properties (chain conditions, generic types) via superstable groups (Berline-Lascar decomposition and \(\alpha\)-indecomposability) to \(p\)-weight, concentrating on the interaction between groups of finite Morley rank and algebraic groups (see the original review for a detailed discussion of the contents). It assembled results scattered in the literature, often simplified the proofs, and organized everything nicely; it thus became the essential tool for everybody working in the area. Provided they spoke, or at least read, mathematical French.NEWLINENEWLINENEWLINEFourteen years later, this tool is now available to English speakers in the translation by Moses Klein, who already translated Poizat's ``Cours de théorie des modèles'' (1985; Zbl 0583.03001); translated as ``A course in model theory'' (2000; Zbl 0951.03002). Again, there are plenty of Gallicisms: What sounds strange in English just has to be translated back into French (but then you would have read the original anyway\dots), none of them too irritating. And again, as remarked by the author himself in a new preface, the subject has evolved drastically in the meantime; in particular the study of simple groups of finite Morley rank has progressed to the point where a proof of Cherlin's Conjecture, under some tameness hypothesis, seems envisageable, but draws heavily on finite rather than algebraic group theory [see \textit{A. Borovik} and \textit{A. Nesin}, ``Groups of finite Morley rank'' (1994; Zbl 0816.20001)]. This development has not been included in the new edition (it would have meant rewriting the book); instead the author provides a limited update in the form of a two-page postscript and some additional references (mainly on conjectures disproved, and the author's own subsequent contributions).NEWLINENEWLINENEWLINEIf you were unfortunate enough to miss out on the original edition, this one is still a good and enjoyable introduction to the basic theory of stable groups. You will be deprived of the pictures, though...