Symmetric generation of the Rudvalis group

Two symmetric presentations for the Rudvalis sporadic simple group Ru are given, and corresponding presentations in terms of generators and relations are deduced. The first considers the linear group L-4(2) acting imprimitively on 105 letters and so analyses a progenitor of form 2(*105) : L-4(2). Certain short relations follow from the basic lemmas of symmetric generation of groups and the resulting homomorphic image is shown to be Ru. In the second approach we work within Ru to show that the group is generated (uniquely up to conjugation) by a set of seven involutions whose set normalizer in Ru is the linear group L-3(2), and which is such that any set of four of these involutions, no three of which lie on a line of the underlying projective plane, generates a copy of the Tits simple group F-2(4)(2)'. Thus Ru is obtained as an image of the progenitor 2(*7) : L-3(2).