  We prove various properties on the structure of groups whose power graph is
chordal. Nilpotent groups with this property have been classified by Manna,
Cameron and Mehatari [The Electronic Journal of Combinatorics, 2021]. Here we
classify the finite simple groups with chordal power graph, relative to typical
number theoretic oracles. We do so by devising several sufficient conditions
for the existence and non-existence of long cycles in power graphs of finite
groups. We examine other natural group classes, including special linear,
symmetric, generalized dihedral and quaternion groups, and we characterize
direct products with chordal power graph. The classification problem is thereby
reduced to directly indecomposable groups and we further obtain a list of
possible socles. Lastly, we give a general bound on the length of an induced
path in chordal power graphs, providing another potential road to advance the
classification beyond simple groups.
