In the past two decades, there have been far-reaching developments in the problem of determining all finite non-abelian simple groups—so much so, that many people now believe that the solution to the problem is imminent. And now, as I correct these proofs in October 1980, the solution has just been announced. Of course, the solution will have a considerable effect on many related areas, both wi...

The Algebraicity Conjecture treats model-theoretic foundations of algebraic group theory. It states that any simple group of finite Morley rank is an algebraic group over an algebraically closed field. In the mid-1990s a view was consolidated that this project falls into four cases of different flavour: even type, mixed type, odd type, and degenerate type. This book contains a proof of the conj...

Let G be a group and let Autc(G) be the group of central automorphisms of G. We say that a subgroup H of a group G is c-characteristic if α(H) = H for all α ∈ Autc(G). We say that a group G is c-characteristically simple group if it has no non-trivial c-characteristic subgroup. If every subgroup of G is c-characteristic then G is called co-Dedekindian group. In this paper we characterize c-char...

We prove that the simple group L3(5) which has order 372000 is efficient by providing an efficient presentation for it. This leaves one simple group with order less than one million, S4(4) which has order 979200, whose efficiency or otherwise remains to be determined.

In recent years probabilistic methods have proved useful in the solution of several problems concerning finite groups, mainly involving simple groups and permutation groups. In some cases the probabilistic nature of the problem is apparent from its very formulation (see [KL], [GKS], [LiSh1]); but in other cases the use of probability, or counting, is not entirely anticipated by the nature of th...

We construct infinite simple sharply 2-transitive groups. Our result answers an open question of Peter Neumann. In fact, we prove that every group G G </...