Constructing the Harada–Norton group

Details

Authors	John N. Bray and Robert T. Curtis.
Title	Monomial modular representations and symmetric generation of the Harada–Norton group.
Published as	Preprint 2003/02 at the University of Birmingham; J. Algebra 268 (2003), no. 2, 723–743.
Availability	DVI-format or PDF-format (with the known errors corrected).
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.

Known errors / Out-of-date infomation

My address and e-mail address are now out of date. However, as of 15th March 2005, the e-mail address printed in the paper still works. This information has not been updated to the most recent form, as it was correct at the time of publication.
Section 4.1 (page 736 in published paper): C_HN(N⁰) is actually isomorphic to D₁₀ (and so is not equal to ⟨ t ⟩, a cyclic group of order 5). The only elements of order 5 in this subgroup lie in ⟨ t ⟩, so that we still have t^a = t, t², t³ or t⁴, as stated in the paper.
In Section 5, the relation [α, δ] = 1 should be replaced by (αδ)² = 1 in the presentations of HN; they are the same relation in the presentation of HS:2. (This error only affects the preprint as it was spotted in time to correct the proofs of the J.Algebra version.)
The website http://www.mat.bham.ac.uk/spres/ is now http://web.mat.bham.ac.uk/spres/. (So much for stable URLs!) This website is referred to in Section 7 (top of page 741 in published paper) and contains a matrix representation of HN which is also on this page.

Extra information

Some of this work also appears in Chapter 9 of my PhD thesis. My thesis contains some work that did not make it into the paper; the paper also contains some work not in my thesis (because it was done later). The generators used in my thesis for the presentation and representations are different from those in the paper. Representations below are generators used in the paper.

You may find 56 × 56 matrices over Q(√5) generating 2.HS:2 in MAGMA format here. This representation can be reduced (directly) modulo any prime except 5.

You may find 133 × 133 matrices over Q(√5) generating HN in MAGMA format here. This representation can be reduced (directly) modulo any prime except 2 and 5. We recommend that you use a reasonably fast computer if you wish to play with this representation in characteristic 0, otherwise it will take an inordinate amount of time to do anything, like even loading the file. [Actually, matrix multiplication is not that bad, but matrix inversion takes quite a long time; it is this that causes the long loading time.]

The recently made presentations based on our symmetric presentation can be found here and here. (Not the same generating set as used for the matrices above.)

Last updated 16th March 2005
Иван Николай Брей / Джон Нихолас Брей (John Nicholas Bray)