  This article began as a study of the structure of infinite permutation groups
G in which point stabilisers are finite and all infinite normal subgroups are
transitive. That led to two variations. One is the generalisation in which
point stabilisers are merely assumed to satisfy min-N, the minimal condition on
normal subgroups. The groups G are then of two kinds. Either they have a
maximal finite normal subgroup, modulo which they have either one or two
minimal non-trivial normal subgroups, or they have a regular normal subgroup M
which is a divisible abelian p-group of finite rank. In the latter case the
point stabilisers are finite and act irreducibly on a p-adic vector space
associated with M. This leads to our second variation, which is a study of the
finite linear groups that can arise.
