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

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