MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

If you look at the toplogical case, then a profinite group has finite rank if all its closed subgroups are finitely generated (toplogically). Now a pro-$p$ group of finite rank can be characterized in many ways. For instance, a pro-$p$ group has finite rank if and only if it is a Lie group over the $p$-adic integers if and only if it is linear over the $p$-adics if and only if it has polynomial subgroup growth if and only if the associtaed graded Lie algebra (with respect to the Zassenhous-Lazard filtration) is nilpotent.
If you look at the toplogical case, then a profinite group has finite rank if all its closed subgroups are finitely generated (toplogically). Now a pro-$p$ group of finite rank can be characterized in many ways. For instance, a pro-$p$ group has finite rank if and only if it is a Lie group over the $p$-adic integers if and only if it is linear over the $p$-adics if and only if it has polynomial subgroup growth if and only if the associtaed graded Lie algebra (with respect the Zassenhous-Lazard filtration) is nilpotent.