Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-06-24T19:17:32.185Z Has data issue: false hasContentIssue false
  • English
  • Français

An algebraic theory of hypergroups

Published online by Cambridge University Press:  17 April 2009

J.R. McMullen
J.R. McMullen
Affiliation:
Department of Pure Mathematics, University of Sydney, Sydney, New South Wales.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The category of groups forms a full subcategory of the category of hypergroups. This larger category also contains other group theoretic objects, such as the conjugacy class hypergroup and character hypergroup of a finite group.

Definitions of hypergroups and hypergroup morphisms are given and related to double algebras (which are simultaneously algebras and cogebras, but not Hopf algebras in general). Quotient and orbit hypergroups are defined. Coefficients are allowed in a more or less arbitrary field.

These concepts provide a new language in which groups and their character tables can be fruitfully discussed.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 20 , Issue 1 , January 1979 , pp. 35 - 55
Copyright
Copyright © Australian Mathematical Society 1979

References

[1]Brauer, Richard, “On pseudo groups”, J. Math. Soc. Japan 20 (1968), 13–22.CrossRefGoogle Scholar
[2]McMuMllen, J.R. and Price, J.F., “Reversible hypergroups”, Rend. Sem. Mat. Fis. Milano 47 (1977), 67–85CrossRefGoogle Scholar
[3]McMullen, J.R. and Price, J.F., “Duality for finite abelian hypergroups over splitting fields”, Bull. Austral. Math. Soc. 20 (1979), 57–70.CrossRefGoogle Scholar
[4]Pahlings, H., “Characterization of groups by their character tables II”, Comm. Algebra 4 (1976), 111–153.CrossRefGoogle Scholar
[5]Pahlings, H., “Characterization of groups by their character tables II”, Comm. Algebra 4 (1976), 155–178.CrossRefGoogle Scholar
[6]Ross, Kenneth A., “Hypergroups and centers of measure algebras”, Analisi armonica e spazi di funzioni su gruppi localmente compatti, 189–203 (Symposia Mathematica, 22. Academic Press, London and New York, 1977).Google Scholar
[7]Roth, Richard L., “Character and conjugacy class hypergroups of a finite group”, Ann. Math. Pura Appl. 105 (1975), 295–311.CrossRefGoogle Scholar
[8]Spector, René, “Apercu de la théorie des hypergroupes”, Analyse harmonique sur les groupes de Lie, 643–673 (Séminaire Nancy-Strasbourg 1973–75 Lecture Notes in Mathematics, 497. Springer-Verlag, Berlin, Heidelberg, New York, 1975).CrossRefGoogle Scholar
[9]Sweedler, Moss E., Hopf algebras (Benjamin, New York, 1969).Google Scholar