= Rank of a profinite group
{title2=$\operatorname{rk}(G)=\sup_{K\le_cG}d(K)$}
The rank of a <profinite group> is $\operatorname{rk}(G)=\sup_{K\le_cG}d(K)$, where $K$ runs through closed <subgroups> and $d(K)$ is the least number of topological generators. If an <open subgroup> $H$ has rank $r$, its <normal core> $C$ is open and has rank at most $r$. Every closed $K\le G$ has a <normal subgroup> $K\cap C$ generated by at most $r$ elements, while $K/(K\cap C)$ has order at most $[G:C]$. A <finite group> of order at most $M$ is generated by at most $\lfloor\log_2M\rfloor$ elements, since each necessary new generator at least doubles the generated <subgroup>. Lifting generators yields $d(K)\le r+\lfloor\log_2[G:C]\rfloor$. Thus finite rank is preserved in both directions under passage to an <open subgroup>.
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