# What's “projective” about “projective pro-finite groups”?

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 $G$.
I think you mean any continuous group epimorphism from it to a profinite quotient group $G/H$ lifts to a continuous group homomorphism to the profinite group $G$. Are you trying to talk about a similarity with projective modules? If so, we should explain why it is an epimorphism we're talking about and not a homomorphism. –  Makhalan Duff Nov 2 '10 at 13:16