= Hereditarily just-infinite profinite group
= Hereditarily just infinite
{synonym}
An infinite <profinite group> is <hereditarily just infinite> if every <open subgroup> is a <just-infinite profinite group>. To prove this, it suffices to show that every nonidentity element of every <open subgroup> has open closed <normal closure>: any nontrivial closed <normal subgroup> then contains that <normal closure>. The <Nottingham group> is an example. In odd characteristic its depth-commutator formula constructs nonzero elements of each sufficiently high layer inside any such <normal subgroup>, and successive coefficient correction fills the entire tail.
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