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\(C^*\)-algebras and the classification of finite groups. (English) Zbl 0736.20014

Operator theory, operator algebras and applications, Proc. Summer Res. Inst., Durham/NH (USA) 1988, Proc. Symp. Pure Math. 51, Pt. 2, 345-354 (1990).
[For the entire collection see Zbl 0699.00028.]
This paper is partly expository with a sprinkling of new results and ideas. Our goal is to achieve a deeper understanding of finite groups. To this end, we perhaps raise more questions than we answer. One of the broad, even vague, questions which motivated the following discussion is, “Why is the number of groups of order \(n\) such a seemingly chaotic function of \(n\)?” One of our goals, not yet fully achieved, is to “see” in some geometric way “why” there are the number of groups of order \(n\) that there are. We hope that new insights into the structure of finite groups will be a product of this investigation. This paper is a written (and expanded) version of a talk given at the 36th Summer Research Institute of the American Mathematical Society, Durham, New Hamshire, July 3-23, 1988. We include here some results discovered since the Summer Research Institute.

MSC:

20D60 Arithmetic and combinatorial problems involving abstract finite groups
46L05 General theory of \(C^*\)-algebras
11P81 Elementary theory of partitions
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
05A17 Combinatorial aspects of partitions of integers
11N45 Asymptotic results on counting functions for algebraic and topological structures

Citations:

Zbl 0699.00028
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