A profinite group is said to be projective if its cohomological dimension is $\leq 1$. Is this related to some other notion of "projective"? How so?
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A profinite group $P$ is projective if and only if any continuous group homomorphism from it to a profinite quotient group $G/H$ lifts to a continuous group homomorphism to the profinite group $G$. 


It is explained in Section 7.6 of Ribes and Zalesskii, "Profinite groups". The notion is similar to the notion of a projective module. For example, free profinite groups are projective. Moreover, a profinite group is projective if and only if it is a closed subgroup of a free profinite group. 

