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Generators of Monothetic Groups

Published online by Cambridge University Press:  20 November 2018

D. L. Armacost
D. L. Armacost*
Affiliation:
Amherst College, Amherst, Massachusetts
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A topological group G is called monothetic if it contains a dense cyclic subgroup. An element x of G is called a generator of G if x generates a dense cyclic subgroup of G. We denote by E(G) the set of generators of G; the complement of E(G) in G, consisting of the “non-generators” of G, we write as N(G) Throughout this paper we consider only locally compact abelian (LCA) groups satisfying the T2 separation axiom (note that a monothetic group is automatically abelian). In [1] certain problems of measurability concerning the set E(G) are discussed. In this paper we shall consider some algebraic and topological properties of the sets E(G) and N(G)

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 5 , October 1971 , pp. 791 - 796
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Halmos, P. and Samelson, H., On monothetic groups, Proc. Nat. Acad. Sci. U.S.A. 28 (1942), 254/258.Google Scholar
2. Hewitt, E. and Ross, K., Abstract harmonic analysis, Vol. I (Academic Press, New York, 1963).Google Scholar
3. Kaplansky, I., Infinite abelian groups (University of Michigan Press, Ann Arbor, 1954).Google Scholar