So, by the symmetry of a system, we mean a set of operators, each of which commutes with the Hamiltonian, and they form a symmetry group. This group may be abelian or non-abelian. Depending upon which one it is, the properties of the system changes (for example, if the group is abelian, there would be no degeneracy). Corresponding to every different kind of symmetry in a system, we can find a symmetry group associated with it.