On the uniqueness of the Harada-Norton group

Abstract

The definition of a group 2 + ’ + 8. (A 5 wr2)-type is given in (2.2.1). See also (2.3.1) and the notation in (2.1.4) and (2.1.5) for (2 .HS): 2. A group G satisfying the hypotheses of the main theorem exists. Its existence is a consequence of the existence of the Monster, constructed by R. Griess in 151. We know of no published work proving the uniqueness of F5. Of course, many properties of F, were derived in Harada’s major work on F5 [7]. A group G satisfying the hypotheses of the main theorem will be said to be of F,-type. Let G be a group of F,-type and let a, z be involutions in G as in the Main Theorem. We consider the graph f defined as follows. Its vertex set is the conjugacy class a . G Two involutions x, y E aG form an edge iff xy~a~. We show that the hypotheses of the Main Theorem imply the uniqueness of r and that G = [Aut(T), Aut(T)] and hence G is unique. Our paper relies on some properties of (2 . HS): 2 (see (2.3)) and some properties of co(z), G, z as in the Main Theorem (but we do not assume, but rather derive, the isomorphism type of c&)). Otherwise our paper is independent of [7] and we prove all properties of G which we require. After some preliminary (and important) work in Section 2, we start in Section 3 to obtain properties of G and r (r as above). Perhaps the most important result of Section 3 is that C,(a) has 8 nontrivial orbits on aG as given in the following table: 261 0021-8693192 55.00