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Groups arise in nature as “sets of symmetries (of an object), which are closed under composition and under taking inverses”. For example, the symmetric group Sn is the group of all permutations (symmetries) of {1,...,n}; the alternating group A n is the set of all symmetries preserving the parity of the number of ordered pairs (did you really remember that one?); the dihedral group D2n is the group of symmetries of the regular n-gon in the plane.

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