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groups

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Reminder about groups On this page, will denote a general binary operation on a (nonempty) set G . Recall that a binary operation takes two inputs (in a specific order) from G and produces an output which must also be in G . In other words, the expression “binary operation” will, in this class, automatically include the so-called closure condition: ( g, h G ) g h G. Definition. Let G be a (nonempty) set, and let be a binary operation on G . We say that ( G, ) is a group if the following three conditions are satisfied. (i) (associativity) ( g h ) k = g ( h k ) for all g, h, k G . (ii) (existence of identity) There is an e G such that e g = g e = g for all g G . [it is easy to see that this e is unique] (iii) (existence of inverse) For every g G there is an h G such that g h = h g = e . The number of elements in G (cardinality of G ) is called the order of the group: o ( G ) = | G | . If the group satisfies the additional property ( g, h G ) g h = h g , then it is said to be commutative or Abelian . Remarks. When ( G, ) is a group, we often say that G is a group under (or with respect to) the operation , or simply say that G is a group. Typically, a multiplicative notation is used by writing “ · ” for the operation . In this case we sometimes write 1 for the identity e , and g - 1 for the inverse of g . We also often drop the symbol · altogether, and simply write gh for g · h , and g 2 , g 3 , etc, for repeated “multiplications.”

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