  Two elements $g$ and $h$ of a permutation group $G$ acting on a set $V$ are
said to be intersecting if $g(v) = h(v)$ for some $v \in V$. More generally, a
subset ${\cal F}$ of $G$ is an intersecting set if every pair of elements of
${\cal F}$ is intersecting. The intersection density $\rho(G)$ of a transitive
permutation group $G$ is the maximum value of the quotient $|{\cal F}|/|G_v|$
where ${\cal F}$ runs over all intersecting sets in $G$ and $G_v$ is a
stabilizer of $v\in V$. In this paper the intersection density of transitive
groups of degree twice a prime is determined, and proved to be either $1$ or
$2$. In addition, it is proved that the intersection density of transitive
groups of prime power degree is $1$.
