# Algebra - Abstract Algebra

### set operation integers axioms

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.

See also Associative property; Solution of equation.

## Resources

### Books

Bittinger, Marvin L, and Davic Ellenbogen. *Intermediate Algebra: Concepts and Applications.* 6th ed. Reading, MA: Addison-Wesley Publishing, 2001.

Blitzer, Robert. *Algebra and Trigonometry.* 2nd ed. Englewood Cliffs, NJ: Prentice Hall, 2003.

Gelfond, A.O. *Transcendental and Algebraic Numbers.* Dover Publications, 2003.

Immergut, Brita and Jean Burr Smith. *Arithmetic and Algebra* *Again.* New York: McGraw-Hill, 1994.

Stedall, Jacqueline and Timothy Edward Ward. *The Greate Invention of Algebra: Thomas Harriot's Treatise on Equations.* Oxford: Oxford University Press, 2003.

Weisstein, Eric W. *The CRC Concise Encyclopedia of Mathematics.* New York: CRC Press, 1998.

### Other

*Algebra Blaster 3 CD-ROM for Windows.* Torrance, CA: Davidson and Associates Inc., 1994.

J.R. Maddocks

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