Jajte introduced the operator semi-stable distributions on Rⁿ in [2] and proved an important fact: A full distribution μ is operator semi-stable, if and only if, there exist a number c(0 < c < 1), a vector h ∈ Rⁿ, and a nonsingular linear operator B in Rⁿ such that the formula μ^c = Bμ*δ(h) holds. In this paper, we make use of the eigenvalue of the matrix B to give a necessary and sufficient condition for ∫_|x|≤1|x|^rM(dx) < ∞, where M is the Lvy measure of μ. Also, we use the symmetric group of μ to characterize the operators B in (1).