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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).
##### MSC:
 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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