
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 (â€˜group theoryâ€™) has turned out to be one of the most fertile areas of 19th and 20thcentury 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 a1, such that a + b = e. (The element b is known as the inverse of a, and takes the place ofâ€”a in the integers.) SMcL
See also category theory; fields; Galois theory; Hilbert\'s problems; rings. 
