Abstract
Given a finite group G, its double Burnside ring B(G,G), has a natural duality operation that arises from considering opposite (G,G)-bisets. In this article, we systematically study the subgroup of units of B(G,G), where elements are inverse to their dual, so called orthogonal units. We show the existence of an inflation map that embeds the group of orthogonal units of B(G/N,G/N) into the group of orthogonal units of B(G,G), when N is a normal subgroup of G, and study some properties and consequences. In particular, we use these maps to determine the orthogonal units of B(G,G), when G is a cyclic p-group, and p is an odd prime.
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Acknowledgements
The author would like to thank the referee for their helpful suggestions. Particularly, in suggesting a much more natural proof of Theorem 4.6.
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Presented by: Pramod Achar
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Barsotti, J. Orthogonal Units of the Double Burnside Ring. Algebr Represent Theor 23, 1683–1705 (2020). https://doi.org/10.1007/s10468-019-09894-4
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DOI: https://doi.org/10.1007/s10468-019-09894-4