  We discuss various symmetry properties of the reparametrization invariant toy
model of a free non-relativistic particle and show that its commutativity and
noncommutativity (NC) properties are the artifact of the underlying symmetry
transformations. For the case of the symmetry transformations corresponding to
the noncommutative geometry, the mass parameter of the toy model turns out to
be noncommutative in nature. By exploiting the Becchi-Rouet-Stora-Tyutin (BRST)
symmetry transformations, we demonstrate the existence of the NC and show its
cohomological equivalence with its commutative counterpart. A connection
between the usual gauge symmetry transformations corresponding to the
commutative geometry and the quantum groups, defined on the phase space, is
also established for the present model at the level of Poisson bracket
structure. We show that, for the NC geometry, such a kind of quantum group
connection does not exist.
