Constructing the Harada–Norton group


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.
AvailabilityDVI-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 3A7, the exceptional triple cover of the alternating group A7. In this paper, a 5-modular monomial representation of 2HS: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Known errors / Out-of-date infomation

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)