  For a permutation group $G$ acting on a set $V$, a subset $\mathcal{F}$ of
$G$ is said to be an intersecting set if for every pair of elements $g,h\in
\mathcal{F}$ there exists $v \in V$ such that $g(v) = h(v)$. The intersection
density $\rho(G)$ of a transitive permutation group $G$ is the maximum value of
the quotient $|\mathcal{F}|/|G_v|$ where $G_v$ is a stabilizer of a point $v\in
V$ and $\mathcal{F}$ runs over all intersecting sets in $G$.
  If $G_v$ is a largest intersecting set in $G$ then $G$ is said to have the
Erd\H{o}s-Ko-Rado (EKR)-property. This paper is devoted to the study of
transitive permutation groups, with point stabilizers of prime order with a
special emphasis given to orders 2 and 3, which do not have the EKR-property.
Among other, constructions of infinite family of transitive permutation groups
having point stabilizer of order $3$ with intersection density $4/3$ and of
infinite families of transitive permutation groups having point stabilizer of
order $3$ with arbitrarily large intersection density are given.
