  Noncommutative quantum mechanics on the plane has been widely studied in the
literature. Here, we consider the problem using Isham's canonical group
quantization scheme for which the primary object is the symmetry group that
underlies the phase space. The noncommutativity of the configuration space
coordinates requires us to introduce the noncommutative term in the symplectic
structure of the system. This modified symplectic structure will modify the
group acting on the configuration space from abelian $\mathbb{R}^2$ to a
nonabelian one. As a result, the canonical group obtained is a deformed
Heisenberg group and the canonical commutation relation (CCR) corresponds to
what is usually found in noncommutative quantum mechanics.
