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

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.