Introduction

A group is called simple if it has no non-trivial normal subgroups. Just as prime numbers are the building blocks of number theory, simple groups play the same role for groups. Take for example a group ${\sl\large G}$ with a normal subgroup ${\sl\large G_1}$ of largest order. Then the factor group ${\sl\large G}/{\sl\large G_1}$ is simple. This process can be iterated until we have what is called a Composition Series

$$1 = {\sl\large G_0} \le {\sl\large G_1} \le {\sl \large G_2} ... ..... \le {\sl\large G_{k-1}} \le {\sl\large G_k} = {\sl\large G}$$

where each factor group ${\sl\large G_{i+1}}/{\sl\large G_i}$ is simple.

The Jordan-Holder Theorem in the early 1860's showed that every group has a composition series. Because many of a groups properties can be determined by the nature of its composition series, the study of simple groups has great importance for the study of groups in general. This is the motivation for the Holder Program, whose first goal was the clasification of all finite simple groups.

The first simple group was discovered by Galois in the late 1830's. In searching for a solution to the quintic equation, Galois used the simplicity of the permutation group A₅ to show that there is no solution by radicals to the quintic.

More and more simple groups were discovered. The next leap was the four infinte families of simple matrix groups discovered by Jordan in 1870. Progress grew slowly until the massive result of the Feit-Thompson Theorem in the early 1960's, taking 255 pages to show that a non-abelian simple group must have even order. Between 1966 and 1975, 19 more sporadic simple groups were discovered, but it was thought that the problem would not be solved until the year 2000. In 1980, a simple group was discovered by Robert Greiss. It was a group of rotations in 196,883 dimensions! So every element can be represented as a 196,883 X 196,883 matrix. Because of its gigantic order

808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000

it is rightfully called the "monster."

Not soon after this discovery, Daniel Gorenstein announced that all finite simple groups had been discovered. The proof of what is called the "Enormous Theorem" is contained in 15,000 journal pages, with hundreds of individual papers, and efforts of over 100 mathematicians.

Theorem: There is a list consisting of 18 infinite familes of simple groups and 26 sporadic simple groups not belonging to these families such that every finite simple group is isomorphic to one of the groups in the list.

The date was January 1981, almost twenty years before the anticipated solution. And maybe more surprising is the fact that there are so few non-abelian simple groups, being only five with order less than 1000, and 56 with order less than 1 million. Here, we state some useful theorems and apply them to classify all the simple groups of order between 200 and 400. The end result demonstrating that the only simple groups in this range are the abelian groups of prime power order, and the group of even permutations A₆, whose order equals 360.