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.   Let  be an operad defined over a field of characteristic zero. Let R be a cogroup in the category of complete -algebras. In this article, we show that R is necessarily the completion of a free -algebra. We also handle the case of cogroups in connected graded algebras over an operad, and the case of groups in connected graded coalgebras over an operad.

Cogroups in algebras over an operad are free algebras