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$K$-theory spectra at height 
$p-1$
Using the descent spectral sequence for a Galois extension of ring spectra, we compute the Picard group of the higher real 
$K$-theory spectra of Hopkins and Miller at height 
$n=p-1$, for 
$p$ an odd prime. More generally, we determine the Picard groups of the homotopy fixed points spectra 
$E_{n}^{hG}$, where 
$E_{n}$ is Lubin–Tate 
$E$-theory at the prime 
$p$ and height 
$n=p-1$, and 
$G$ is any finite subgroup of the extended Morava stabilizer group. We find that these Picard groups are always cyclic, generated by the suspension.
