Vol. 80, No. 2, 1979

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Double coset and orbit spaces

David Kent Harrison

Vol. 80 (1979), No. 2, 451–491

It is our purpose to study abstractly and in general, the structure of the set of all double cosets of a group with respect to a subgroup. In our flrst section we allow the groups to be infinite, and focus on a ternary collineation relation. If X, Y , Z are double cosets, we say they are collinear if there exists x X, y Y , z Z with x y z = 1. This relation is abstracted and forms the basis of that section. In the second section we essentially insist the groups are finite, and count the double cosets which products from two cosets can appear in (i.e., count all Z with x y Z for x X, y Y ). We abstract the properties that these numbers possess, and study their implications. The orbit space of conjugacy classes of a finite group can be taken as a set of double cosets (in the holomorph of the group). This set has a natural dual, and in certain instances other sets of double cosets have one also. We study this situation in the third and last section by use of a complex matrix which enjoys some of the properties of the character table of a finite group; this may be thought of as another approach (slightly different than Brauer’s pseudogroups) to character tables as a thing in themselves.

Mathematical Subject Classification 2000
Primary: 51D25
Secondary: 20N99, 22E99, 51D30
Received: 31 August 1977
Revised: 2 August 1978
Published: 1 February 1979
David Kent Harrison