  Hexagonal LuFeO$_3$ has drawn a lot of research attention due to its
contentious room-temperature multiferroicity. Due to the unstability of
hexagonal phase in the bulk form, most experimental studies focused on
LuFeO$_3$ thin films which can be stabilized by strain using proper substrates.
Here we report on the hexagonal phase stabilization, magnetism, and
magnetoelectric coupling of bulk LuFeO$_3$ by partial Sc-substitution of Lu.
First, our first-principles calculations show that the hexagonal structure can
be stabilized by partial Sc substitution, while the multiferroic properties
including the noncollinear magnetic order and geometric ferroelectricity remain
robustly unaffected. Therefore, Lu$_{1-x}$Sc$_x$FeO$_3$ can act as a platform
to check the multiferroicity of LuFeO$_3$ and related materials in the bulk
form. Second, the magnetic characterizations on bulk Lu$_{1-x}$Sc$_x$FeO$_3$
demonstrate a magnetic anomaly (probable antiferromagnetic ordering) above room
temperature, $\sim425-445$ K, followed by magnetic transitions in low
temperatures ($\sim167-172$ K). In addition, a magnetoelectric response is
observed in the low temperature region. Our study provides useful information
on the multiferroic physics of hexagonal $R$FeO$_3$ and related systems.
