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Existentially closed groups. (English) Zbl 0646.20001

W. R. Scott, in 1951 [see Proc. Am. Math. Soc. 2, 118-121 (1951; Zbl 0043.023)] defined “algebraically closed” and “weakly algebraically closed” groups: If every finite set of equations and inequalities over a group G that can be solved in some supergroup of G can be solved in G itself, then G is algebraically closed; and weakly algebraically closed if this is demanded for finite sets of equations only. W. R. Scott there asked whether every non-trivial weakly algebraically closed group is algebraically closed. This was shown to be the case by the reviewer [J. Lond. Math. Soc. 27, 247-249 (1952; Zbl 0046.248)]. Then interest in the subject died. It came to life again in 1969, with a lecture the reviewer delivered at Irvine, California [see Stud. Logic Found. Math. 71, 553-562 (1973; Zbl 0262.20046)], and since then many workers in group theory, recursion theory, model theory, and other fields have obtained many deep and interesting results. There has been a change in the nomenclature: what Scott called “weakly algebraically closed” is now called “algebraically closed”, and his “algebraically closed” has become “existentially closed”. By now there is an extensive literature on existentially closed groups and other existentially closed algebraical structures.
The book under review makes no attempt at being comprehensive: it grew out of a course of lectures Professor Higman gave at Oxford in 1983-84, “intended to be of interest to any mathematician with an interest in group theory or logic, while also being accessible to first-year graduate students. The book reflects this aim.” Nevertheless the book covers much ground: it starts with group theory, goes on to recursion theory, games, and finally the first-order theory of existentially closed groups. The approach is largely informal, but it is made quite clear how rigid formality could be injected. This is a delightfully user-friendly book, with plenty of explanation of what is going on and why. It is produced to the high standards of the Clarendon Press; a few misprints will keep the reader on her toes.
Reviewer: B.H.Neumann

MSC:

20-02 Research exposition (monographs, survey articles) pertaining to group theory
20E34 General structure theorems for groups
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20A15 Applications of logic to group theory
03D80 Applications of computability and recursion theory
03C60 Model-theoretic algebra
20F05 Generators, relations, and presentations of groups
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