Robert T. Curtis

Symmetric representation of the elements of the Janko group J 1

Abstract In this paper we represent each element of the Janko group J 1 as a permutation on eleven letters from the projective special linear group L 2 (11), followed by a word of length at most four in the eleven involutory symmetric generators. An algorithm for multiplying elements represented in this way is described, and computer programs which carry out this procedure and others are given in the appendix.

A systematic approach to symmetric presentations. I. Involutory generators

In this paper we conduct a systematic, computerized search for groups generated by small, but highly symmetric, sets of involutions. Many classical groups are readily obtained in this way, as are a number of sporadic simple groups. The techniques of symmetric generation developed elsewhere are described afresh, and the results are presented in a convenient tabular form.

Monomial modular representations and symmetric generation of the Harada–Norton group

Abstract This paper is a sequel to Curtis [J. Algebra 184 (1996) 1205–1227], where the Held group was constructed using a 7-modular monomial representation of 3·A7, the exceptional triple cover of the alternating group A7. In this paper, a 5-modular monomial representation of 2·HS:2, a double cover of the automorphism group of the Higman–Sims group, is used to build an infinite semi-direct product P which has HN, the Harada–Norton group, as a ‘natural’ image. This approach assists us in constructing a 133-dimensional representation of HN over Q ( 5 ) , which is the smallest degree of a ‘true’ characteristic 0 representation of P . Thus an investigation of the low degree representations of P produces HN. As in the Held case, extension to the automorphism group of HN follows easily.

On graphs and codes

In a recent paper we showed how the binary Golay code can be obtained in a revealing way straight from the edge-graph of the icosahedron. This construction not only yields a natural basis for the code, but also supplies a simple description of all codewords. In this paper we show that the above is merely a special case of a general method of constructing codes from graphs. Codes with certain properties, such as self-orthogonality, can be obtained by putting certain conditions on the graph with which we start.

Symmetric presentations for the Fischer groups I: the classical groups Sp6(2), Sp8(2), and 3·O7(3)

Abstract We give computer-free proofs for symmetric presentations of the groups Sp 6 (2), Sp 8 (2), and 3· O 7 (3).

A systematic approach to symmetric presentations II: Generators of order 3

In this paper we conduct a systematic computerized search for groups generated by small, but highly symmetric, sets of elements of order 3. Many classical groups are readily obtained in this way, as are a number of sporadic simple groups. Firstly, we introduce monomial modular representations as these will prove useful later in the paper. Then the techniques of symmetric generation developed elsewhere are described afresh. The results we obtain are presented in a convenient tabular form, together with relevant character tables.

Symmetric Generation and Existence of McL : 2, the Automorphism Group of the McLaughlin Group

We use the primitive action of the Mathieu group M22 of degree 672 to define a free product of 672 copies of the cyclic group ℤ2 extended by M22 to form a semidirect product which we denote by P = 2☆672: M 22. Such a semidirect product is called a progenitor. By investigating a subprogenitor of shape 2☆42: A 7 we are led to a short relation by which to factor P. We verify that the resulting factor group is McL: 2, the automorphism group of the McLaughlin simple group, and identify it with the familiar permutation group of degree 275.

The Leech Lattice Lambda and the Conway group .O revisited

We give a new, concise definition of the Conway group ·O as follows. The Mathieu group M 24 acts quintuply transitively on 24 letters and so acts transitively (but imprimitively) on the set of ( 24 4 ) tetrads. We use this action to define a progenitor P of shape 2 * ( 24 4 ): M 24 ; that is, a free product of cyclic groups of order 2 extended by a group of permutations of the involutory generators. A simple lemma leads us directly to an easily described, short relator, and factoring P by this relator results in ·O. Consideration of the lowest dimension in which ·O can act faithfully produces Conways elements ξ T and the 24―dimensional real, orthogonal representation. The Leech lattice is obtained as the set of images under ·O of the integral vectors in R 24 .

The Thompson chain of subgroups of\break the Conway~group Co1 and complete graphs on

Abstract The large Conway simple group Co 1

{n}

{{\rm Co}_{1}}