= Procyclic group
= Procyclic
{synonym}
A <profinite group> $G$ is procyclic if every quotient by an <open normal subgroup> is cyclic. Equivalently $G=\overline{\langle g\rangle}$ for one element $g$. For the forward implication, the sets of lifts of generators of $G/N$ are nonempty <clopen> sets. Any finite intersection contains a lift of a generator of the cyclic quotient by the intersection of those <open normal subgroups>. <Compactness> supplies one element generating every finite quotient, and hence generating densely. The reverse implication follows by projecting that generator. For a <pro-p group>, two distinct maximal <open subgroups> would give a quotient $C_p^2$, so a nontrivial <procyclic group> has a unique maximal <open subgroup> $M$. Its <profinite Frattini subgroup> is $M$, and any $g\notin M$ generates topologically: otherwise the proper <closed subgroup> it generates lies in a maximal <open subgroup>. The additive <group> of <p-adic integers> is an infinite example, generated topologically by $1$.
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