  If G is a connected linear algebraic group over the field k, a Levi factor of
G is a reductive complement to the unipotent radical of G. If k has positive
characteristic, G may have no Levi factor, or G may have Levi factors which are
not geometrically conjugate. We give in this paper some sufficient conditions
for the existence and the conjugacy of Levi factors of G.
  Let A be a Henselian discrete valuation ring with fractions K and with
perfect residue field k of characteristic p>0. Let G be a connected and
reductive algebraic group over K. Bruhat and Tits have associated to G certain
smooth A-group schemes P whose generic fibers P/K coincide with G; these are
known as *parahoric group schemes*. The special fiber P/k of a parahoric group
scheme is a linear algebraic group over k. If G splits over an unramified
extension of K, we show that P/k has a Levi factor, and that any two Levi
factors of P/k are geometrically conjugate.
