@GWOMaths One might say that physicists study the symmetry of nature, while mathematicians study the nature of symmetry.

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Observing symmetry in nature, such as noting the similarity between the symmetries of a snowflake and a hexagon, is readily comprehensible. What does it mean then to study "the nature of symmetry"?

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Mathematicians define a "group", G, as a set of elements {a,b,c, ...} with a binary operation ⊡ and a distinguished element e (the identity of G) satisfying these specific properties:

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For all a, b, c in G
1) Closure: a⊡b is in G.
2) Associativity: (a⊡b)⊡c = a⊡(b⊡c)
3) Identity: e⊡a = a⊡e = a.
4) Inverse: There exists an element a* such that a*⊡a = a⊡a* = e.

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Just as functions on the integers, rationals, or reals are defined as mappings, mathematicians define a *group morphism* μ as a mapping from one group to another that preserves the group structure:

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