On algebraically closed groups

In this paper we use model-theoretic techniques to analyze the structure of algebraically closed groups. The notion of algebraically closed group first appeared in W. R. Scott's paper [24] in 1951. The intention must surely have been to provide for grouptheory an analogue of that central notion of field theory, the notion of algebraically closed field. A group G is said to be algebraically closed if every consistent finite system of equations, with parameters in G, is solvable in G. A system of equations is said to be consistent over G if it has a solution in a group extending G. A group G is said to be existentially closed if every consistent finite system of equations and inequations, with parameters in G, is solvable in G.

[1]  C. C. Chang.  Some remarks on the model theory of infinitary languages , 1968 .

[2]  Erwin Engeler,et al.  Languages with expressions of infinite length , 1966 .

[3]  Abraham Robinson,et al.  Completing theories by forcing , 1970 .

[4]  G. Higman,et al.  Embedding Theorems for Groups , 1949 .

[5]  B. H. Neumann,et al.  A Note on Algebraically Closed Groups , 1952 .

[6]  R. Vaught,et al.  Homogeneous Universal Models , 1962 .

[7]  J. L. Britton,et al.  THE WORD PROBLEM , 1963 .

[8]  K. Jon Barwise,et al.  Infinitary logic and admissible sets , 1969, Journal of Symbolic Logic.

[9]  G. Higman.  Subgroups of finitely presented groups , 1961, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[10]  W. R. Scott.  Algebraically closed groups , 1951 .

[11]  Paul C. Eklof,et al.  Model-completions and modules , 1971 .

[12]  H. Simmons,et al.  The Word Problem for Absolute Presentations , 1973 .

[13]  Angus Macintyre,et al.  Omitting quantifier-free types in generic structures , 1972, Journal of Symbolic Logic.