Profinite completion

Summary

The profinite completion of a group $G$ is the inverse limit of all its finite quotient groups. It is a profinite group that records every finite quotient of $G$ at once, together with a natural homomorphism from $G$. This homomorphism is injective exactly when $G$ is residually finite. The profinite completion of $\Z$ is the group $\hat\Z$ of profinite integers, which is the product of the $p$-adic integers over all primes $p$.