University of Birmingham > Talks@bham > Algebra Seminar > Symmetric generation of the Rudvalis group

Symmetric generation of the Rudvalis group

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If you have a question about this talk, please contact Simon Goodwin.

A free product of n copies of the cyclic group of order m, which we denote by m*n, possesses many monomial automorphisms which permute the n generators and raise them to powers prime to m. Indeed the group of all such automorphisms is soon seen to have order phi(m)n x n!, where phi(m) denotes the number of natural numbers less than m and co-prime to it. If N is a subgroup of this group which acts transitively on the n cyclic subgroups then a semi-direct product of form P = m*n:N is called a progenitor. Symmetric generation of groups is concerned with finding interesting finite images of such progenitors.

It turns out that the case m=2 is particularly fruitful and many symmetric presentations of sporadic simple groups have been found needing just one short additional relation. Several of these will be mentioned briefly by way of motivation, but the the body of the talk will be devoted to the Rudvalis simple group Ru which proved surprisingly resistant to this approach.

In the work we shall describe we take N=L4(2) acting naturally on a 4-dimensional vector space over Z2. It turns out that if we apply a simple lemma, which restricts the form of relators by which it is sensible to factor P, we are led directly to defining relators for Ru.

This talk is part of the Algebra Seminar series.

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