  Consider the algebra Q<<x_1,x_2,...>> of formal power series in countably
many noncommuting variables over the rationals. The subalgebra Pi(x_1,x_2,...)
of symmetric functions in noncommuting variables consists of all elements
invariant under permutation of the variables and of bounded degree. We develop
a theory of such functions analogous to the ordinary theory of symmetric
functions. In particular, we define analogs of the monomial, power sum,
elementary, complete homogeneous, and Schur symmetric functions as will as
investigating their properties.
