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.
Historical articles
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)
Hermann Weyl, Symmetry, Journal of the Washington Academy of Sciences 28 6 (1938) 253-271 jstor:24530200
On symmetry and introducing the language of homotopy type theory for univalent foundations of mathematics:
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