The universal embedding for the involution geometry of \(Co_1\)
type: Item
instance of: scholarly article
title: The universal embedding for the involution geometry of \(Co_1\)
zbMATH Open document ID: 0940.51010
author: Matthew Kirby Bardoe
published in: Journal of Algebra
publication date: 2000-07-17T00:00:00Z
review text: The author proves that the universal embedding for the \(2\)-local involution geometry for \(Co_1\) over \(\text{GF}(2)\) is 300-dimensional. This geometry is described in [\textit{M. A. Ronan} and \textit{S. D. Smith}, Proc. Symp. Pure Math. 37, 283-289 (1980; Zbl 0478.20015)]. In two previous papers the author investigated the universal embedding for the Suzuki sporadic simple group and for the unitary group \(\text{U}_4(3)\), where \(\text{U}_4(3) \subset \text{Suz} \subset Co_1\) [see \textit{M. K. Bardoe}, J. Algebra 186, No. 2, 367-383 (1996; Zbl 0868.51014) and ibid., 447-460 (1996; Zbl 0868.51015)]. The dimensions of the universal embeddings of \(\text{U}_4(3)\) and of Suz are 70 and 143, respectively. An embedding is an incidence preserving map of an abstract geometry into a Pappian projective space \(P\) such that the points and lines of this geometry are mapped injectively to points and lines of \(P\), respectively. For \(2\)-local involution geometries there exists a universal embedding, i.e. an embedding which can be mapped surjectively onto every other embedding of that geometry. The proof of the dimension theorem for \(Co_1\) is based on information about certain subgeometries of \(Co_1\), namely the involution geometry for \(G_2(2)\) (the dual hexagon), the involution geometry for \(J_2\) (the near-octagon geometry), and the Suzuki involution geometry [see \textit{W. M. Kantor}, Eur. J. Comb. 2, 239-247 (1981; Zbl 0514.51006), \textit{D. E. Frohardt} and \textit{S. D. Smith}, Eur. J. Comb. 13, No. 6, 455-472 (1992; Zbl 0802.51008) and \textit{M. K. Bardoe}, loc. cit.].
Mathematics Subject Classification ID: 51E24
Mathematics Subject Classification ID: 20D08
zbMATH DE Number: 1340560
zbMATH Keywords: universal embedding
zbMATH Keywords: involution geometry
describes a project that uses: GAP
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1340560