The Simple Group J_{4}: Summary of Ph.D. Thesis

D. J. Benson

Cambridge, 1981

Thesis advisor: Professor J. G. Thompson

One of the important problems in group theory is the classification of the finite simple groups. In Autumn 1980 it was announced that the proof of the classification theorem is now complete. There are certain known infinite families, and 26 sporadic groups. One of these, namely J_{4}, was constructed in Cambridge last year by Norton, Parker, Conway, Thackray and myself. My thesis describes this joint work culminating in the proof of the existence of J_{4}, and some work of my own arising out of this.

Chapter 1 is preparatory: it describes notations for working with the Mathieu groups. These groups are involved in the 2-local subgroups of J_{4}, and in Chapters 2–5 notations are developed for working inside an ‘incident’ set of maximal 2-local subgroups of J_{4}. Chapter 6 describes the computer construction of J_{4} via a 112-dimensional representation over the field of two elements, and gives the proof that the group constructed is indeed isomorphic to J_{4}. The proof is an interesting blend of abstract group-theoretic arguments and practical computer work. The main heavy computer work in the proof is involved in showing that the skew-square of the 112-dimensional representation has an invariant subspace of dimension 4995. Chapter 7 gives further details of the internal geometry of the 112-dimensional representation, and finds a presentation for J_{4} by generators and relators. This presentation consists of adjoining two relators to the amalgamated product of the 2-local subgroups M of shape 2^{11}M_{24} and H of shape 2^{1+12}3M_{22}.2. One of the relators comes from a maximal 2-local subgroup P of shape 2^{3+12}(S_{5} × L_{3}(2)) and the other involves a subgroup isomorphic to P GL_{2}(23) intersecting M in L_{2}(23). In an appendix, generators are given for the suborbit stabilizers for the actions of M and H in the primitive permutation action of J_{4} on the 173, 067, 389 cosets of M.