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Symmetry in mathematics

From Wikipedia, the free encyclopedia

Symmetry occurs not only in geometry, but also in other branches of mathematics. Symmetry is a type of invariance: the property that a mathematical object remains unchanged under a set of operations or transformations.[1]

The root system of the exceptional Lie group E8. Lie groups have many symmetries.

Given a structured object X of any sort, a symmetry is a mapping of the object onto itself which preserves the structure. This can occur in many ways; for example, if X is a set with no additional structure, a symmetry is a bijective map from the set to itself, giving rise to permutation groups. If the object X is a set of points in the plane with its metric structure or any other metric space, a symmetry is a bijection of the set to itself which preserves the distance between each pair of points (i.e., an isometry).

In general, every kind of structure in mathematics will have its own kind of symmetry, many of which are listed in the given points mentioned above.