= Discrete remnant of a circle symmetry
{title2=$H=\mathbb Z_{\gcd(Q_1,\ldots,Q_r)}$}
Suppose fields with integer <U(1) charges> $Q_i$ acquire nonzero <vacuum expectation values>, with at least one $Q_i\ne0$. The <stabilizer> in the <circle group> is exactly $\mathbb Z_d$, where $d=\gcd(|Q_1|,\ldots,|Q_r|)$. Indeed, $e^{i\theta}$ preserves every condensate if and only if $e^{iQ_i\theta}=1$ for every $i$. The <Bezout identity> implies $e^{id\theta}=1$, and conversely $d\mid Q_i$ proves every $d$th <root of unity> preserves every condensate. Neutral fields impose no constraint; if all condensing fields are neutral, the whole <circle group> survives.
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