  One-dimensional hole gas confined in a cylindrical Ge nanowire has potential
applications in quantum information technologies. Here, we analytically study
the low-energy properties of this one-dimensional hole gas. The subbands of the
hole gas are two-fold degenerate. The low-energy subband wave-functions are
obtained exactly, and the degenerate pairs are related to each other via a
combination of the time-reversal and the spin-rotation transformations. In
evaluating the effective $g$-factor of these low-energy subbands, we find the
orbital effects of the magnetic field are as important as the Zeeman term.
Also, near the center of the $k_{z}$ space, there is a sharp dip or a sharp
peak structure in the effective $g$-factor. At the site $k_{z}=0$, the
longitudinal $g$-factor $g_{l}$ is much less than the transverse $g$-factor
$g_{t}$ for the lowest subband, while away from site $k_{z}=0$, $g_{l}$ can be
comparable to $g_{t}$.
