Elementary abelian \(\varvec{p}\)-groups are the only finite groups with the Borsuk–Ulam property

  • Ikumitsu NagasakiEmail author


It is well known that the Borsuk–Ulam theorem holds for elementary abelian p-groups \(C_p{}^k\). When the Borsuk–Ulam theorem holds for a finite group G, we say that G has the Borsuk–Ulam property or G is a BU-group. In this paper, we show that a non-abelian p-group of exponent p is not a BU-group, which leads to a complete classification of finite BU-groups, namely finite BU-groups are only elementary abelian p-groups.


Borsuk–Ulam theorem Borsuk–Ulam property BU-group representation sphere equivariant map 

Mathematics Subject Classification

Primary 55M20 Secondary 57S17 



The author would like to thank the referee for carefully reading the manuscript and for giving valuable comments and suggestions.


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Authors and Affiliations

  1. 1.Department of MathematicsKyoto Prefectural University of MedicineKyotoJapan

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