On Ree’s series of simple groups

Abstract

Ree recently discovered a series of finite simple groups related to the simple Lie algebra of type (G2) [5; 6 ] . We have determined the irreducible characters of these groups. In this work, we do not use the actual definition of Ree's groups, but only the properties (l)-(5) given below. Since these are sufficient to determine the bulk of the character tables of these groups, it is our hope that they actually characterize the groups completely. The five properties used are (1) If G is one of these groups, the order of G is even and the 2Sylow subgroup of G is elementary Abelian of order 8. (2) G has no normal subgroup of index 2. (3) There is an involution / (an element of order 2) in G such that if C(J) is the centralizer of / in G and (J ) is the group generated by / , then C(J)/(J) is isomorphic to LF(2, q), the linear fractional group in two variables over a field of q elements. Here we restrict q by q^3 (mod 8) and q^27. I t follows that C(J) is the direct product of (ƒ) and a subgroup F of G which is isomorphic to LF(2, q). In F there is an element R of order (q — l)/2f and we also have (4) If R5*l, then C(R)QC(J). Finally, there is an element S of F with order (q+l)/2. Let J' = S* where / = ( g + l ) / 4 . Then S generates the commutator subgroup of C(J, the centralizer of (J , J'). We have (5) F o r S 2 , C(S*)QC(J). This condition can actually be replaced by the weaker condition (5*) Let A €N(J, J'), the normalizer in G of (ƒ, / '>, but A $ C(J, J)Let A = 1. Then A does not commute with S. From these conditions on G we derive a number of results. The approach is almost entirely by means of characters, both ordinary and modular. Condition (4) leads to two families of exceptional characters related to the classes of R( 5^1) and JR (T^J). These characters are the characters of 2-defect 1. Results of Brauer [ l ; 2; 3] then lead to two possibilities for the principal 2-block (the 2-block containing the character which is 1 everywhere) ; one contains seven 1 This note is a summary of the results obtained in the author's Harvard University dissertation of the same title, written under the direction of Richard Brauer. 2 This research was supported by the National Science Foundation.