The multiplicators of certain simple groups

Abstract

The recent paper of Steinberg [7] on the multiplicators of the finite simple groups of type, the classical determination of the multiplicators of the alternating groups by Schur [6], a similar result of Janko for his group [3] and the (unpublished) work of J. G. Thompson on the Mathieu groups cover all but three families of known simple groups. In this paper we give a simple determination of the multiplicators for two of these families, namely the Suzuki groups and the Ree groups of characteristic three. Our results are well known for the Suzuki groups, with the exception of the one of smallest order, while the determination for the Ree groups of characteristic three has been accomplished by J. H. Walter with the use of some deep theorems of modular character theory. However, our main tool is an elementary lemma of Lie type involving a very crude numerical estimate. In addition, we calculate the multiplicator for the smallest Suzuki group. Furthermore, preliminary investigations indicate that our methods might show that the multiplicators, or at least their 2-primary components, are trivial for the remaining family of Ree groups of characteristic two defined over GF(22n+') for all n > N, where N is fairly small. Our lemma deals with a single automorphism; a suitable generalization of this result to a pair of commuting automorphisms would suffice to prove the preceding statement. Our main results are: