= Symmetry enlargement under group translates
{title2=$S=\bigcup_{g\in G}g(T)$}
For a <group> of <Euclidean isometries>, every element preserves the displayed union of images of a <set>, because left multiplication permutes its <group> indices. Its full setwise <isometry group> can be larger even when $T$ itself has no nonidentity symmetry. Take $G$ to be the <integer> <translations> of the plane and $T=\{(0,0),(1,0),(0,2)\}$. By <rigidity of a scalene triangle>, $T$ has trivial <stabilizer subgroup>. Yet $S=\mathbb Z^2$, whose quarter-turn about the origin is a symmetry outside the <translation> <group>.
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