Group
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A group is a set of elements together with an operation
(the dot is frequently supressed) satisfying the following conditions:
- For all
,
(associativity).
- There exists an element
so that for all
,
(identity).
- For any
, there exists
so that
( inverses).
Groups frequently arise as permutations 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
(since it has
elements). Another example of a group is the symmetric group
of all permutations of
.