  A transitive permutation group $G$ on a finite set $\Omega$ is said to be
pre-primitive if every $G$-invariant partition of $\Omega$ is the orbit
partition of a subgroup of $G$. It follows that pre-primitivity and
quasiprimitivity are logically independent (there are groups satisfying one but
not the other) and their conjunction is equivalent to primitivity. Indeed, part
of the motivation for studying pre-primitivity is to investigate the gap
between primitivity and quasiprimitivity. We investigate the pre-primitivity of
various classes of transitive groups including groups with regular normal
subgroups, direct and wreath products, and diagonal groups. In the course of
this investigation, we describe all $G$-invariant partitions for various
classes of permutation groups $G$. We also look briefly at conditions similarly
related to other pairs of conditions, including transitivity and
quasiprimitivity, $k$-homogeneity and $k$-transitivity, and primitivity and
synchronization.
