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Groups of height four

Published online by Cambridge University Press:  09 April 2009

Alfred W. Hales
Alfred W. Hales
Affiliation:
Department of Mathematics University of California, Los Angeles, U.S.A.
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Abstract

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If G and H are infinite groups then G is said to be larger than H (H≼G) if there are subgroups A of G, B of H, each of finite index, such that B is an epimorphic image of A. Pride (1979) showed that if G has finite ‘height’ with respect to the quasi-order ≼ then there are only finitely many (classes of) minimal groups H with H ≼G, and asked whether this were true without the minimality restriction on H. This paper gives a negative answer to his question by exhibiting a group G of height four with infinitely many (classes of) groups H satisfying H≼G.

1980 Mathematics subject classification (Amer. Math. Soc.): 20 E 99, 20 K 15.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 29 , Issue 3 , May 1980 , pp. 297 - 300
Copyright
Copyright © Australian Mathematical Society 1980

References

Beaumont, R. A. and Pierce, R. S. (1961), ‘Torsion-free groups of rank two’, Mem. Amer. Math Soc. 38.Google Scholar
Fuchs, L. (1970, 1973), Infinite abelian groups, Vols. I and II (Academic Press, New York).Google Scholar
Kaplansky, I. (1969), Infinite abelian groups (University of Michigan Press, Ann Arbor, Michigan).Google Scholar
Pride, S. J. (1976), ‘The concept of “largeness” in group theory’, Proc. Conf. on Word and Decision Problems in Algebra and Group Theory, Oxford, 1976 (to appear).Google Scholar
Pride, S. J. (1979), ‘On groups of finite height’, J. Austral. Math. Soc. (Series A) 28, 87–99.CrossRefGoogle Scholar