We clarify several subtle points in the CP N-1 model in 2 + 1 dimensions. It is shown that as a consequence of local gauge invariance the nA particles, belonging to fundamental representation of SU(N) do not appear in the spectrum (in any number of dimensions) regardless of the nature of the interaction between them. In 2 + 1 dimensions we analyse the phase structure of the theory from the point of view of realization of the relevant symmetries: SU(N)/ZN ⊗ UΦ(1) ⊗ UE(1). The symmetries UΦ(1) and UE(1) are gauge invariant analogs of the magnetic flux and electric charge symmetries in scalar QED. The charge of UE(1) is not related to the global part of the local U(1) transformations (which counts the difference between the number of n and the number of n* excitations). At the phase transition point the flavour SU(N) ZN is broken down to U(N-1) while the mode of implementation of the UΦ(1) is changed from Kosterlitz-Thouless to Wigner-Weyl. We provide a gauge invariant order parameter for the flavour symmetry breaking and identify the massless "photon" with the Kosterlitz-Thouless zero mode of UΦ(1).