- For all , (associativity).
- There exists an element so that for all , (identity).
- For any , there exists so that ( inverses).
(Equivalently, a group is a monoid with inverses.)
Groups frequently arise as permutations or symmetries of collections of objects. For example, the rigid motions of that fix a certain regular -gon is a group, called the dihedral group and denoted in some texts (since it has elements) and in others (since it preserves a regular -gon). Another example of a group is the symmetric group of all permutations of .
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