Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-25T19:37:08.310Z Has data issue: false hasContentIssue false
  • English
  • Français

Characteristically simple ℵ0-categorical groups

Published online by Cambridge University Press:  12 March 2014

Robert H. Gilman
Robert H. Gilman*
Affiliation:
Stevens Institute of Technology, Hoboken, New Jersey 07030
Get access Rights & Permissions [Opens in a new window]

Abstract

A countable group is ℵ0-categorical if it is characterized up to isomorphism among all countable groups by its first order theory. We investigate countable ℵ0-categorical groups which are characteristically simple and have proper subgroups of finite index. We determine such groups up to isomorphism (Corollary 4 to Proposition 3), and we show that their theories are finitely axiomatizable (Proposition 2).

Let K be a finite nonabelian simple group. In §3 we construct from K two denumberable groups, K* and K#, and use them to classify countable direct limits of finite Cartesian powers of K (Proposition 1). The results of §3 are used in §4 to show that K* and K# are ℵ0-categorical and to find axioms for their theories. By Corollary 2 to Proposition 2, K* and K# are counterexamples to the conjecture of U. Feigner [3, p. 309] that ℵ0-categorical groups are FC-solvable. Also in §4 the results mentioned in the first paragraph are proved. §2 is devoted to preliminary lemmas about finite Cartesian powers of K.

A characteristic subgroup of a group G is a subgroup which is mapped onto itself by all automorphisms of G. G is characteristically simple if its only characteristic subgroups are itself and the identity subgroup. Each characteristic subgroup of a group is a union of orbits of the automorphism group of the group. If G is ℵ0-categorical, then as the automorphism group of G has only a finite number of orbits, G has only a finite number of characteristic subgroups.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 49 , Issue 3 , September 1984 , pp. 900 - 907
Copyright
Copyright © Association for Symbolic Logic 1984

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Baur, W., Cherlin, G. and Macintyre, A., Totally categorical groups and rings, Journal of Algebra, vol. 57 (1979), pp. 407–440.CrossRefGoogle Scholar
[2]Chang, C. C. and Keisler, H. J., Model theory, 2nd ed., North-Holland, Amsterdam, 1977.Google Scholar
[3]Felgner, U., Stability and ℵ0-categoricity of nonabelian groups, Logic Colloquium '76 (Gandy, R. O. and Hyland, J. M. E., editors), North-Holland, Amsterdam, 1977, pp. 301–324.Google Scholar
[4]Huppert, B., Endliche Gruppen. I, Springer-Verlag, Berlin, 1967.CrossRefGoogle Scholar
[5]Macintyre, A., Model completeness, Handbook of mathematical logic (Barwise, J., editor), North-Holland, Amsterdam, 1977, pp. 139–180.CrossRefGoogle Scholar
[6]Robinson, A., Infinite forcing in model theory, Proceedings of the Second Scandinavian Logic Symposium (Oslo, 1970), North-Holland, Amsterdam, 1971, pp. 317–340.CrossRefGoogle Scholar
[7]Rosenstein, J., ℵ0-categoricity of groups, Journal of Algebra, vol. 25 (1973), pp. 435–467.CrossRefGoogle Scholar
[8]Rosenstein, J., ℵ0-categoricity is not inherited by factor groups, Algebra Universalis, vol. 6 (1976), pp. 93–95.CrossRefGoogle Scholar
[9]Saracino, D., Model companions for ℵ0-categorical theories, Proceedings of the American Mathematical Society, vol. 39 (1973), pp. 591–598.Google Scholar
[10]Scott, W. R., Group theory, Prentice-Hall, Englewood Cliffs, New Jersey, 1964.Google Scholar
[11]Waszkiewicz, J. and Wȩglorz, B., On ω0-categoricity of powers, Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 17 (1969), pp. 195–199.Google Scholar
[12]Wilson, J. S., The algebraic structure of ℵ0-categorical groups, Groups—St. Andrews, 1981 (Campbell, C. M. and Robertson, E. F., editors), London Mathematical Society Lecture Notes Series, vol. 71, Cambridge University Press, Cambridge, 1982, pp. 345–358.CrossRefGoogle Scholar
[13]Apps, A. B., Boolean powers of groups, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 91 (1982), pp. 375–395.CrossRefGoogle Scholar
[14]Apps, A. B., On the structure of ℵ0-categorical groups (to appear).Google Scholar