= Singlet criterion for an unbroken subgroup
{title2=$V^H\ne\{0\}$}
An unbroken <subgroup> $H$ must fix the <vacuum expectation value> $v$ of a <scalar field> in a <group representation> $V$. Thus a nonzero such $v$ exists exactly when $V^H\ne\{0\}$. For a <compact group>, <complete reducibility of compact-group representations> identifies this fixed <vector subspace> with the sum of the <trivial representations> in the <restriction of a representation> $V|_H$. Every $v\in V^H$ has $H\subseteq\operatorname{Stab}(v)$; equality requires a separate <stabilizer> calculation. The criterion neither determines the full unbroken <group> nor guarantees that a given <scalar potential> has its minimum at that direction.
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