  Single-component quantum gas confined in a harmonic potential, but otherwise
isolated, is considered. From the invariance of the system of the gas under a
displacement-type transformation, it is shown that the center of mass
oscillates along a classical trajectory of a harmonic oscillator. It is also
shown that this harmonic motion of the center has, in fact, been implied by
Kohn's theorem. If there is no interaction between the atoms of the gas, the
system in a time-independent isotropic potential of frequency $\nu_c$ is
invariant under a squeeze-type unitary transformation, which gives collective
{\it radial} breathing motion of frequency $2\nu_c$ to the gas. The amplitudes
of the oscillating and breathing motions from the {\it exact} invariances could
be arbitrarily large. For a Fermi system, appearance of $2\nu_c$ mode of the
large breathing motion indicates that there is no interaction between the
atoms, except for a possible long-range interaction through the
inverse-square-type potential.
