A symmetry is (in most classical situations) invariance under a group action, or infinitesimally, invariance under a Lie algebra action. Indeed, historically the mathematical term “group” is a contraction of group of symmetries (e.g. Klein 1872).

In physics, see local symmetry, global symmetry, spontaneous symmetry breaking.


  • Felix Klein, Vergleichende Betrachtungen über neuere geometrische Forschungen (1872)

    translation by M. W. Haskell, A comparative review of recent researches in geometry , trans. M. W. Haskell, Bull. New York Math. Soc. 2, (1892-1893), 215-249. (retyped pdf, retyped pdf, scan of original)

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