In a much quoted article
MANIFOLD attributed the
rising popularity of
simple groups to fashion.
This longstanding assertion
is challenged in:

complex simplicity
Chris Rowley

From the slights and sneers heaped upon finite group theory in general, and the study of simple groups in particular it seems that a "Group Theorists Apology" along the lines of Hardy would be now, more than ever, appropriate; but readers must look elsewhere for such a defence. This article pretends to defend only one part of group theory, the study of simple groups, on the counts of relevance to the whole of group theory, and their fascinating character (c).

A simple group is a group whose only normal subgroups are the subgroup comprising only the identity and the group itself (the so called trivial subgroups). These groups are the building blocks from which all finite groups are made up. Technically any finite group G has a composition series; i.e. a series

{1} Ì H₁ Ì H₂ Ì ...... Ì H_n-1 Ì H_n = G

such that H_i D H_i+1 (each is a normal subgroup of the next) and the factor (coset ) group H_i+l/H_i is a simple group for i = 1, 2, ... n - 1.

When we have a list of all the simple groups we need then to examine how they can be fitted together to form new groups, through a composition series, then we can list all finite groups. These methods of manufacturing new groups are called group extensions - and this is another part of finite group theory; but it is the first task which is occupying most attention at present.

The simplest of simple groups are the groups of prime order, which, since the order of a subgroup must divide the order of the group, have no non-trivial subgroups, hence no normal ones. Groups which are built up of these groups are called soluble (or solvable by Americans to distinguish them from Copper Sulphate). This does not mean that everything is known about these groups, on the contrary the study of them is still one of the most fruitfull branches of the subject; but it cannot produce any new simple groups, because the simple groups appearing in the composition series of a soluble group must be cyclic of prize order.

Apart from this family finite simple groups have a very complicated structure and the problem of finding them all is nowhere near solved. Many conditions are known which a group must satisfy to stand a chance of being simple, but there are much stronger conditions which are satisfied by all the known ones. Is this because there are lots of finite groups which have not yet been discovered. or because, although we cannot yet prove. it, all simple groups must satisfy these stronger conditions? The answer is not yet clear but probably the truth lies somewhere in between the extremes.

I shall now describe some of the types of property which are being investigated in order to classify the finite simple groups. It was conjectured by Burnside, at the beginning of this century that all finite simple groups, except for the cyclic ones mentioned earlier, have even order. This was proved by Walter Feit and John Thompson in 1963, in an epic 255 page paper called "The Solvability of Groups of Odd Order", this paper must also have broken all records for the theorem with the longest proof. From this result we can show that any simple group contains an involution, (an element which is of order two, i.e. is its own inverse). In fact we prove that any group of even order contains an involution. It is a group axiom that every element has an inverse, thus we pair off each element with its inverse. This leaves us with the identity, which is its own inverse, but the group has even order so there must be at least one other element left out of this pairing, that is at least one involution. Since non-cyclic finite simple groups are certainly not soluble they must have even order and hence contain involutions. It is the centraliser of an involution in such a group which is of interest to the group theorist. (The centraliser of an element of a group is the set of all elements in the group which commute with the element i.e. if x is the element then the centraliser of x consists of all y for which xy = yx). This subgroup is non-trivial since it contains the element itself, and it cannot be the whole group since if it were the subgroup {1, x} would be a non-trivial normal subgroup contradicting the group's simplicity.

Many,workers, notably Daniel Gorenstein and Zvanimir Janko, are at present engaged in trying to pin down all finite simple groups by first proving that the subgroups occuring in them as centralisers of involutions can only have certain forms and then finding all groups which contain involutions with centralisers of these forms. This would classify all simple groups, and is the method of attack which is proving most successful at present.

A generalisation of this is to consider the Sylow p subgroups of a simple group. (Sylow proved that every group G has a subgroup with p^r elements in it where p is any prime, and r is the maximal power of p dividing the order of G.)

Another method is to find all the simple groups which enjoy the property that all their proper subgroups are of a certain form. An example of this is Thompson's gargantuan work in which he finds (all) the finite simple groups, all of whose proper subgroups are soluble (these are called N-groups). This paper has a rough history: it was thought to be complete when, about three years ago, Hearne pointed out that a group known as Tits' group was an N-group. This was unfortunate because it was not included in Thompson's list, though he claimed to have proved that his list included all such groups. Obviously something was wrong. Mistakes were found but they could not be patched up simply by the addition of Tits' group to the list; this led to a period of confusion during which rumours spread that Thompson had given up trying to patch up his proof, together with counter~rumours that it had been completed. The latest news is that it has now been finished with only Tits' group added to the original list. I am sure this has caused many group theorists to breath a sigh of relief as many recent results have relied on Thompson's result for their proof.

The ultimate theorem of this nature which we should like to be able to prove is what is called the K-group theorem. A K-group is defined to be any known finite simple group and the theorem would be of the form:- Let G be a finite simple group such that any proper subgroup of G has a composition series, each of whose factors are K groups, then G is a K-group. This theorem would only be true when the set of K-groups includes all finite simple groups; as its truth implies that we have found all of them. At the moment this seems a very distant goal as new finite simple groups are being discovered which do not seem to fit in to any pattern at all. Thus the set of K-groups grows bigger yearly and may well continue to do so for some time yet. Whether or not the methods used at present are the right ones, and will result in a complete cladsification they are certainly providing a lot of knowledge about the structure of these groups and what makes them simple

BIBLIOGRAPHY
Simple Groups: Eve La Chyl, MANIFOLD-2
J. Thompson, Nice 1970, MANIFOLD-8