  In this paper we study the group $A_0(X)$ of zero dimensional cycles of
degree 0 modulo rational equivalence on a projective homogeneous algebraic
variety $X$. To do this we translate rational equivalence of 0-cycles on a
projective variety into R-equivalence on symmetric powers of the variety. For
certain homogeneous varieties, we then relate these symmetric powers to moduli
spaces of \'etale subalgebras of central simple algebras which we construct.
This allows us to show $A_0(X) = 0$ for certain classes of homogeneous
varieties, extending previous results of Swan / Karpenko, of Merkurjev, and of
Panin.
