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We study generalised P and CP transformations in the three-Higgs-doubletmodel (3HDM) with Higgs and gauge fields only. We find that there are twoequivalence classes, with respect to flavour transformations, of generalised Ptransformations and there is only one class of CP transformations. We discussthe conditions the potential has to satisfy in order to be invariant underthese transformations. We apply the method of bilinears which we brieflyreview. We discuss the relation to the conventional basis, where the potentialis written in terms of scalar products of the doublet fields. In particular wereproduce the known result that a potential is invariant under CPtransformations if and only if there is a conventional basis where allparameters are real. Eventually we study standard P and CP transformations inthe $n$-Higgs-doublet model (nHDM). We show that for the bilinears of the nHDMthe standard CP transformation corresponds to a diagonal linear transformationwith only $\pm 1$ as diagonal elements. We give this matrix explicitly forarbitrary $n$.

Generalised P and CP transformations in the 3-Higgs-doublet model