Abstract
LetG be a compact group of automorphism acting continuously on a compact groupH. Then the orbit spaceH G is a compact hypergroup. We characterize, all solvable groupsH and compact automorphism groupsG for whichH G is almost discrete, i.e.,H G is homeomorphic to the one-point-compactification of ℕ. It turns out that thenH is isomorphic either to the infinite direct product ℤ(p)ℕ of the cyclic groups ℤ(p) or to ℕ n p (ℤ p the group of allp-adic numbers) for some primep and some ℕ. The almost discrete orbit hypergroupsH G are determined explicitly for some examples.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Connett, W. C., Schwartz, A. L.: Product formulas, hypergroups and the Jacobi polynomials. Bull. Amer. Math. Soc.22, 91–96 (1990).
Dickson, L. E.: Cyclotomy, higher congruences, and Waring's problem. Amer J. Math.57, 391–424 (1935).
Dunkl, C. F., Ramirez, D. E.: A family of countable compact P*-hypergroups. Trans. Amer. Math. Soc.202, 339–356 (1975).
Hewitt, E., Ross, K. A.: Abstract Harmonic Analysis I. Berlin-Heidelberg-New York: Springer. 1979.
Jacobson, N.: Totally disconnected locally compact rings. Amer. J. Math.58, 433–449 (1936).
Jewett, R. I.: Spaces with an abstract convolution of measures. Adv. in Math.18, 1–101 (1975).
Morris, S. A., Oates-Williams, S., Thompson, H. B.: Locally compact groups with every closed subgroup of finite index. Bull. London Math. Soc.22, 359–361 (1990).
Schwartz, A. L.: Classification of one-dimensional hypergroups. Proc. Amer. Math. Soc.103, 1073–1081 (1988).
Voit, M.: Central limit theorems for random walks on0 that are associated with orthogonal polynomials. J. Multiv. Analysis34, 290–322 (1990).
Voit, M.: Projective and inductive limits of hypergroups. Proc. London Math. Soc.67, 617–648 (1993).
Voit, M.: Substitution of open subhypergroups. Hokkaido Math. J.23, 143–183 (1994).
Voit, M.: Compact almost discrete hypergroups. Can. J. Math.48, 201–214 (1996).
Vrem, R.: Hypergroup joins and their dual objects. Pacific J. Math.111, 483–495 (1984).
Vrem, R.: Connectivity and supernormality results for hypergroups. Math. Z.195, 419–428 (1987).
Weil, A.: Basic Number Theory. Berlin-Heidelberg-New York: Spriger 1967.
Wildberger, N. J.: Hypergroups and cyclotomy. University of New South Wales. Preprint 1990.
Wildberger, N. J.: On the algebraic structure of Gaussian periods. University of New South Wales. Preprint 1991.
Zeuner, H.: One-dimensional hypergroups. Adv. in Math.76, 1–18 (1989).
Zeuner, H.: Duality of commutative hypergroups. In: Probability Measures, on Groups X. Proc. Conf., Oberwolfach, 1990, 467–488. Plenum Press. 1992.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Voit, M. Compact groups having almost discrete orbit hypergroups. Monatshefte für Mathematik 122, 239–250 (1996). https://doi.org/10.1007/BF01320187
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01320187