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Examples are given of profinite groups that are not strongly complete, andhave other `bad' properties, yet have only finitely many open subgroups of eachfinite index. It is shown that a profinite group with the latter property mustbe finite if it has finite exponent. The problem of characterizing stronglycomplete groups in terms of their power subgroups is discussed.

Remarks on profinite groups having few open subgroups