A group is a simple mathematical system, so basic that groups appear wherever one looks in mathematics. Despite the primitive nature of a group, mathematicians have developed a rich theory about them. Specifically, a group is a mathematical system consisting of a set G and a binary operation * which has the following properties:
- x*y is in G whenever x and y are in G (closure).
- (x*y)*z = x*(y*z) for all x, y, and z in G (associative property).
- There exists and element, e, in G such that e*x=x*e=x for all x in G (existence of an identity element).
- For any element x in G, there exists an element y such that x*y=y*x=e (existence of inverses).
Note that commutativity is not required. That is, it need not be true that x*y=y*x for all x and y in G.
One example of a group is the set of integers, under the binary operation of addition. Here the sum of any two integers is certainly an integer, 0 is the identity, -a is the inverse of a, and addition is certainly an associative operation. Another example is the set of positive fractions, m/n, under multiplication. The product of any two positive fractions is again a positive fraction, the identity element is 1 (which is equal to 1/1), the inverse of m/n is n/m, and, again, multiplication is an associative operation.
The two examples we have just given are examples of commutative groups. (Also known as Abelian groups in honor of Niels Henrik Abel, a Norwegian mathematician who was one of the early users of group theory.) For an example of a non-commutative group consider the permutations on the three letters a, b, and c. All six of them can be described by
I is the identity; it sends a into a, b into b, and c into c. P then sends a into a, b into c, and c into b. Q sends a into b, b into a, and c into c and so on. Then P*Q=R since P sends a into a and Q then sends that a into b. Likewise P sends b into c and Q then sends that c into c. Finally, P sends c into b and Q then sends that b into a. That is the effect of first applying P and then Q is the same as R.
Following the same procedure, we find that Q*P=S which demonstrates that this group is not commutative. A complete "multiplication" table is as follows:
From the fact that I appears just once in each row and column we see that each element has an inverse. Associativity is less obvious but can be checked. (Actually, the very nature of permutations allows us to check associativity more easily.) Among each group there are subgroups-subsets of the group which themselves form a group. Thus, for example, the set consisting of I and P is a subgroup since P*P=I. Similarly, I and T form a subgroup.
Another important concept of group theory is that of isomorphism. For example, the set of permutations on three letters is isomorphic to the set of symmetries of an equilateral triangle. The concept of isomorphism occurs in many places in mathematics and is extremely useful in that it enables us to show that some seemingly different systems are basically the same.
The term "group" was first introduced by the French mathematician Evariste Galois in 1830. His work was inspired by a proof by Abel that the general equation of the fifth degree is not solvable by radicals.
Bell, E.T. Men of Mathematics. Simon and Schuster, 1961.
Grossman, Israel, and Wilhelm Magnus. Groups and Their Graphs. Mathematical Association of America, 1965.