Abstract algebra is a branch of mathematics which builds upon set theory. A set is a collection of elements in which order is unimportant. Set theorists also consider the nature of the elements in the set irrelevant; the rules governing set operations are the same, regardless of whether the elements in the set are people, numbers, words, possible outcomes of an event, or something else. However, in abstract algebra, the elements of a set will often be numbers. If the nature of the elements of the set is unspecified, they will usually be referred to by letters.
A course in abstract algebra will usually begin with group theory. A group is a set with a defined operation- let’s call that operation “*”- that is closed under that operation and has the properties of associativity and invertibility. Also, a group must contain a special element called the identity element. The word “closed” refers to the fact that performing an operation on two elements of the set results in another element of that set.
For example, the set of all real numbers is a group under the operation known as multiplication. Multiply any two real numbers, and the result is still a real number; hence, the real numbers are closed under multiplication. The property of associativity holds, since multiplying a string of real numbers will give the same result regardless of how you group those numbers. Multiplying three and four, then multiplying the result by five will yield the same result as multiplying five and four, then multiplying the result by three.
The set of real numbers also has a special “identity” element. An identity element does not change another element when used as an argument in the operation. The number one is the identity element for the real numbers since multiplying any number by one gives you the same number you started with. The number maintains its identity when multiplied by one. Hence, one is the identity element.
Finally, the real numbers satisfy the property of invertibility, since each real number has a multiplicative inverse. Two elements are multiplicative inverses of each other if multiplying them together gives the identity element (one). For example, three and one third are multiplicative inverses.
After learning these basic properties of groups, the student of abstract algebra will proceed to learn additional theorems regarding groups, as well as gaining insight into special types of groups such as cyclic groups, symmetry groups, and topological groups. Other structures such as rings and ideals are studied as well.
Abstract algebra provides the logical foundations for geometry, matrix algebra, and other practical fields of study. Many students in engineering or computer science will find they need to take a course in abstract algebra to complete their graduate studies. Other fields of mathematics such as probability theory, analysis, and combinatorics rely on abstract algebra as well. It is one of the cornerstones of modern mathematics.
