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Let G be a finite group, and let V and W be finite-dimensional real orthogonal G-modules with V 3 W, and with unit spheres S(V) and S( W) respectively. The purpose of this note is to give necessary sufficient conditions for the existence of a C-map /: S( V) -» S( W) in terms of the Burnside ring of G and its relationship with V and W. Note that if W has a nonzero fixed point, such a G-map always exists, so for nontriviality, we assume this not the case.doi:10.1090/s0002-9939-1987-0866449-1 fatcat:2gavsyybgbecpbklt26w2huzxu