  In this paper, we generalize a result of N. Dinculeanu which characterizes
norm compactness in the Bochner space $L^p(G ; B)$ in terms of an approximate
identity and translation operators, where $G$ is a locally compact abelian
group and $B$ is a Banach space. Our characterization includes the case where
$G$ is nonabelian, and we weaken the hypotheses on the approximate identity
used, providing new results even for the case $B = \mathbb{C}$ and $G =
\mathbb{R}^n.$
