  The isotropic harmonic oscillator in N dimensions is shown to have an
underlying symmetry group O(2,1)X O(N)which implies a unique result for the
energy spectrum of the system. Raising and lowering operators analogous to
those of the one-dimensional oscillator are given for each value of the angular
momentum parameter. This allows the construction of an infinite number of
coherent states to be carried out. In the N=1 case there is a twofold family of
coherent states, a particular linear combination of which coincides with the
single set already well known for that case. Wave functions are readily derived
which require only the solution of a first order differential equation, an
attribute generally characteristic of group theoretical approaches.
