ON THE PARTIAL DIFFERENCE SETS IN CAYLEY DERANGEMENT GRAPHS

Abstract

Let $G$ be a finite group. The set $D\subseteq G$with $|D|=k$ is called a $(n,k,\lambda,\mu)$-partial difference set(PDS) in $G$ if the differences $d_1d_2 ^{-1}, d_2,d_2\in  D, d_1\neq d_2$, represent each non-identity element in $D$ exactly $\lambda$  times and each non-identity element in $G-\{D\}$ exactly $\mu$  times.In the present paper, we determine for which group $G\in \{D_{2n},T_{4n},U_{6n},V_{8n}\}$ the derangement set is a PDS. We also prove that the derangement set of a Frobenius group is a PDS.