3.50 (1969)

Let $G$ be a group of order $p^\alpha \cdot m$, where $p$ is a prime, $p$ and $m$ are coprime, and let $k$ be an algebraically closed field of characteristic $p$. Is it true that if the indecomposable projective module corresponding to the 1-representation of $G$ has $k$-dimension $p^\alpha$, then $G$ has a Hall $p'$-subgroup? The converse is trivially true.