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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 nontrivial 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.
