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  A group is one of the most important ideas in abstract algebra. Many structures turn out to be groups, and the investigation of the properties of groups in general (‘group theory’) has turned out to be one of the most fertile areas of 19th- and 20th-century mathematics.

A group consists of a set of some kind, and an operation, which is designed to mimic some of the properties of + on the integers. The operation is a function taking pairs of elements in the set to a third element in the set. The operation satisfies four axioms. (1) If a and b are in the set, then a + b is defined and in the set. (2) If a, b and c are in the set, then (a + (b + c)) = ((a + b) + c). (Brackets show which operations should be performed first; this axiom amounts to saying that the order in which operations are calculated does not matter. This property is known as associativity.) (3) There is an element e in the set such that for every a in the set, a + e = a. (The element e is known as the identity, and mimics the function of 0 in the numbers.) (4) For every element a there is an element b, written a-1, such that a + b = e. (The element b is known as the inverse of a, and takes the place of—a in the integers.) SMcL

See also category theory; fields; Galois theory; Hilbert\'s problems; rings.



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