Infinite degeneracy of Landau levels from the Euclidean symmetry with central extension revisited

The planar Landau system which describes the quantum mechanical motion of a charged particle in plane with uniform magnetic field perpendicular to plane, is explored within pedagogical settings aimed at beginning graduate level. known possess Euclidean symmetry two dimensions central extension $\bar{E}(2)$. In this paper, we revisit well-known energy eigenvalues system, as levels, by exploiting related $\bar{e}(2)$ algebra. Specifically, utilize Casimir operator and commutation relations generators $\bar{E}(2)$ group. More importantly, an algebraic formalism on topic based Schwinger's oscillator model angular momentum also presented. irreducible representations group their implications degeneracy levels discussed.