Subgroups of finite index and the just infinite property

A residually finite (profinite) group G is just infinite if every non-trivial (closed) normal subgroup of G is of finite index. This paper considers the problem of determining which (closed) subgroups of finite index of a just infinite group are themselves just infinite. If G is just infinite and not virtually abelian, we show that G is hereditarily just infinite if and only if all maximal (closed) subgroups of finite index are just infinite. This result will be used to show that a finitely generated pro-p group G is just infinite if and only if G has no non-trivial finite normal subgroups and Φ(G) has a just infinite open subgroup.