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(As Raymond notes, this is not hard to prove for abelian groups G. Also, it is known to be false without the hypothesis of the existence of a stationary point, even for G=Z6; see [3].) Actually, this theorem is a fairly easy consequence of the following result, which was essentially proved by us in [1, Chapter XV, Lemma 2.7] and [2, Lemma 2.2]. It is a generalization of the wellknown fact that the orbits of the action of the Weyl group on the Lie algebra of a maximal torus of a compact Lie group G correspond in a one-one manner to the orbits of the adjoint representation of G with the regular points also corresponding.

The set of nonprincipal orbits of an action on 𝐸ⁿ