Abstract algebra represents a further generalization of elementary algebra. By defining such constructs as groups, based on a set of initial assumptions, called axioms, provides theorems that apply to all sets satisfying the abstract algebra axioms. A group is a set of elements together with a binary operation that satisfies three axioms. Because the binary operation in question may be any of a number of conceivable operations, including the familiar operations of addition, subtraction, multiplication, and division of real numbers, an asterisk or open circle is often used to indicate the operation. The three axioms that a set and its operation must satisfy in order to qualify as a group, are: (1) members of the set obey the associative principle [a × (b × c) = (a × b) × c]; (2) the set has an identity element, I, associated with the operation ×, such that a × I = a; (3) the set contains the inverse of each of its elements, that is, for each a in the set there is an inverse, a', such that a × a' = I. A well known group is the set of integers, together with the operation of addition. If it happens that the commutative principle also holds, then the group is called a commutative group. The group formed by the integers together with the operation of addition is a commutative group, but the set of integers together with the operation of subtraction is not a group, because subtraction of integers is not associative. The set of integers together with the operation of multiplication is a commutative group, but division is not strictly an operation on the integers because it does not always result in another integer, so the integers together with division do not form a group. The set of rational numbers, however, together with the operation of division is a group. The power of abstract algebra derives from its generality. The properties of groups, for instance, apply to any set and operation that satisfy the axioms of a group. It does not matter whether that set contains real numbers, complex numbers, vectors, matrices, functions, or probabilities, to name a few possibilities.
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