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.
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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. |
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