Sporadic groups, code loops and nonvanishing cohomology

Abstract

I intend to discuss a number of interesting 2-locals in sporadic groups and show how code loops, which are certain Moufang loops, may be used to describe the subgroups abstractly. Existence proofs of parabolic subgroups of sporadics which are independent of the existence proofs of the sporadics have not existed in every case. When available, some such demonstrations of existence have been ad hoc. This paper partially alleviates that problem. It is, I believe, the first systematic attempt to describe some of the more complicated parabolics by a unified theme. I was moved to attempt this by Conway’s use of a loop invented by Parker to describe a parabolic of shape 22+ l1 +22 (S, xM~~) in the monster [8]. The first direct construction of this parabolic is due to J. Tits, whose notes (see [35], especially III and IV, and preprint of [37]) were circulated months before Conway’s work was publicized. They may well have influenced Conway’s construction of the monster, though they do not contain the loop concept.