 Let H be a subgroup of G. If for every a G, we have aHa-1 = H, then H is called a normal subgroup of G. Let G1 and G2 be groups. An isomorphism from G1 to G2 is an injection :G1 G2 such that (a · b) = (a) · (b)  (a,b G1). A function f : A B is said to be injective (or one to one) if whenever f(x) = f(y), we have x = y. A function that is both one-to-one and onto, that is if f(x) = f(y) then x = y, and for every y of the domain there is an x of the range so that f(x) = y. A function f : A B is said to be surjective (or onto) if for every y B there exists x A such that f(x) = y.

### Historical Perspectives of the Theory of Groups

At the end of the eighteenth century, Lagrange noticed that if the polynomial

f = Xn + a1Xn-1 + ... + an

with complex coefficients has zeros 1, 2,..., n, then

f = (X - 1)(X - 2)...(X - n),

and therefore,

a1 = -( 1 + ... + n)
a2 = 1 2 + 1 3 + ... + n-1 n
.
.
.
an = (-1)n-1 1 ··· n.

Therefore, Lagrange observed that if 1, ..., n are subjected to any permutation, then the coefficients a1, ..., an are unaltered. To put it another way, the coefficients are symmetric functions of the zeros. It is this observation which is at the heart of Lagrange's method of solving equations of degree < 4 in radicals. Moreover, it is this observation which makes the connection between groups and equations.

Elementary properties of permutations were found by Lagrange, Galois, and Abel, but it was the great mathematician Cauchy who first made an exhaustive study of the elementary properties of permutation groups. Cauchy's work was published in a series of memoirs in the 1840's. It was Cauchy who first clearly enunciated the group concept, although Cauchy studied only groups of permutations. The abstract notion of a group was formulated by Arthur Cayley in 1853. It was Galois who first recognized the importance of normal subgroups of a group in connection with the theory of equations.

The theory of finite groups has been an active field of research for about two centuries, with the main problem being the classification of all finite groups into an organized listing of some form. It is possible to break the classification of all finite groups into two subproblems:

Problem A: Determine all finite simple groups.

Problem B (The Extension Problem): Let H and K be groups. Determine, up to isomorphism, all finite groups G which contain H as a normal subgroup such that G/H K.

Problems A and B are both extremely difficult and no early solution of either is expected in spite of extensive efforts made during the last century.

The problem of determining all finite, simple groups has received a great deal of attention, especially in recent years, when many new examples of simple groups have been discovered.

The work on the extension problem has led to a field of mathematics called the cohomology of groups, which is a field of current interest.

In the latter part of the nineteenth century, a number of mathematicians, in particular Sophus Lie, began to study the class of infinite groups known as continuous groups. In modern terminology, these groups are called Lie Groups. Typical examples of Lie groups are the groups of all translations of of all rigid motions of a plane. The original motivation for studying Lie groups was provided by the theory of differential equations and their applications to mechanics. The theory of Lie groups is an entire discipline in itself and is currently of great importance to the study of particle physics.

Another class of infinite groups is the class of discontinuous groups, which were extensively studied by Poincaré, Klein, and Fricke, as well as many others, in connection with certain functions of a complex variable, called automorphic functions. Today, the theory of automorphic functions is an immense field in which research is currently being done.

Thus, the reader can see that, although the idea of a group was first studied in the rather restricted setting of permutation groups, it has become one of the fundamental unifying concepts of mathematics.