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7.37 (1980)

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We say that a profinite group is strictly complete if each of its subgroups of finite index is open. It is known (B. Hartley, Math. Z., 168, no. 1 (1979), 71–76) that finitely generated profinite groups having a finite series with pronilpotent factors are strictly complete. Is a profinite group strictly complete if it is
$\qquad$ a) finitely generated?
$\qquad$ b) finitely generated and prosoluble?

Contributor: O. V. Mel’nikov

Progress

a) Yes, it is (N. Nikolov, D. Segal, Ann. Math. (2), 165, no. 1 (2007), 171–273).
b) Yes, it is (D. Segal, Proc. London Math. Soc. (3), 81 (2000), 29–54).

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