K-existentially Closed Groups

Date

2024-9-3

Author

Demirci, Caner Doğuş

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For an infinite cardinal κ, a group G is called κ-existentially closed if for every system Σ containing less than κ equations and inequations with constants from G where Σ can be solved in an overgroup H of G, there exists a solution of Σ in G itself. W. R. Scott first defined the κ-existentially closed groups in 1951 in order to generalize algebraically closed fields into group theory. On many occasions in the theory of infinite groups, a specific group called HNN-extension finds itself a place. If a group G has isomorphic subgroups, then an HNN-extension H of G is a group containing G such that the isomorphic subgroups are conjugate in H. We also investigate triangle groups, which are groups T+(l, m, n) for integers l, m, n, possibly infinite and greater than 1, generated by two elements a and b where a has order l, b has order m, and ab has order n. Triangle groups provide a proof of the fact that κ-existentially closed groups are simple groups containing elements of every order. In this paper, our major interest is κ-existentially closed groups following the paper of O. H. Kegel and M. Kuzucuo˘glu with additional attention to HNN-extensions and triangle groups.

Subject Keywords

K-existentially closed groups, Britton’s Lemma, Triangle groups

URI

https://hdl.handle.net/11511/110981

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Graduate School of Natural and Applied Sciences, Thesis