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Free pro-\(p\) groups with operators. (English) Zbl 0755.20006
A semidirect product of a finite \(p\)-group \(P\) by a finite group \(H\) such that the induced representation of \(H\) on the Frattini factorgroup of \(P\) is projective can be extended to the corresponding product of a free pro- \(p\)-group by \(H\) (theorem A). Finitely generated free pro-\(p\)-groups on which \(H\) acts such that the Frattini factorgroups are projective \(\mathbb{F}_ pH\)-modules are projective objects in the category of finitely generated pro-\(p\)-groups with operator group \(H\) (theorem B).
20E18 Limits, profinite groups
20E22 Extensions, wreath products, and other compositions of groups
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
20C25 Projective representations and multipliers
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