The document discusses the classification of quantum systems based on symmetry groups. It shows that a single qubit/qutrit/ququart system is represented by the projective special unitary groups CP1/CP2/CP3 respectively. Larger composite systems like 2-qubits/qutrits/ququarts are represented by higher dimensional projective special unitary groups like CP3/CP8/CP15. In general an N-level quantum system is represented by the projective special unitary group CP2n-1.