Towards the finiteness of π*LK(n)S0

Let G be a closed subgroup of the nth Morava stabilizer group Sn, n⩾2, and let denote the continuous homotopy fixed point spectrum of Devinatz and Hopkins. If G=〈z〉, the subgroup topologically generated by an element z in the p-Sylow subgroup of Sn, and z is non-torsion in the quotient of by its center, we prove that the -homology of any K(n−2)*-acyclic finite spectrum annihilated by p is of essentially finite rank. We also show that the units in En* fixed by z are just the units in the Witt vectors with coefficients in the field of pn elements. If n=2 and p⩾5, we show that, if G is a closed subgroup of not contained in the center, then G contains an open subnormal subgroup U such that the mod(p) homotopy of is of essentially finite rank.