Let $n>3$ be an odd integer and let $K_n$ denote the complete graph on $n$ vertices. For which integers $n$ the line graph $L(K_n)$ is a Cayley graph? For even $n$, it follows from a result of Watkins that $L(K_n)$ is not a Cayley graph. For $n=5$, $L(K_n)$ is the complement of the Petersen graph and so it is not a Cayley graph.