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4 results

  • 15 - The Thompson Chain of Subgroups of Co1

  • Robert T. Curtis, University of Birmingham
  • Book:
    The Art of Working with the Mathieu Group M24
    Published online:
    31 October 2024
    Print publication:
    14 November 2024, pp 234-272

  • Summary

    In the final chapter all the ideas of the book come together to produce the chain of subgroups of the Conway simple group Co1 that was previously referred to as the Suzuki chain. Since this construction emphatically reveals that the chain includes Co1 itself, we prefer to call it the Thompson chain as it was John Thompson who first noted that, with one exception, the normalizers of the groups in the chain are maximal in Co1. In a complete graph on n vertices we let the directed edge from vertex r to vertex s correspond to trs, an element of order 7 in some group where tsr=trs−1. We thus obtain a progenitor of shape

    7⋆(n2):Sn
    in which the symmetric group permutes the vertices. Initially, we include an additional automorphism of the free product that simply squares each of the symmetric generators whilst commuting with the Sn, but we eventually discard it as it is not needed. We must now decide what a triangle 〈t12, t23, t31〉 generates, and we realize that the unitary group U3(3) has all the necessary properties. Factoring by a single relation that ensures that triangles generate copies of this unitary group, we find that a complete 4-graph generates the Hall–Janko group, a complete 5-graph generates the Lie group G2(4), a complete 6-graph generates the triple cover of the Suzuki simple group and a complete 7-graph generates Co1. For n > 7 the group collapses, but if we replace the symmetric group Sn by the alternating group An, then we may proceed as far as n = 9 when a 9-graph also generates Co1. In this configuration a 3-cycle on three vertices lies in the centre of the triple cover of the Suzuki group generated by the edges on the other 6 vertices. We conclude by using MOG techniques to embed this whole configuration into the 24-dimensional representation of the Conway group ·O acting on the Leech lattice, modulo of course the central element of order 2.

  • 14 - Symmetric Generation Using M24

  • Robert T. Curtis, University of Birmingham
  • Book:
    The Art of Working with the Mathieu Group M24
    Published online:
    31 October 2024
    Print publication:
    14 November 2024, pp 209-233

  • Summary

    We describe the method of symmetric generation of groups that has been used to produce pleasing, well-motivated constructions of many of the sporadic simple groups. A progenitor 2*n : N is defined to be a free product of n copies of the cyclic group of order 2, extended by a permutation group N of degree n that permutes these n involutory symmetric generators by conjugation. A useful lemma is introduced that leads to suitable relations by which to factor the infinite progenitor in order to produce a finite image that contains n distinct involutions permuted by an isomorphic copy of N. These ideas are employed using N ≅ M24.

    In the first case, we have n =(244) and the symmetric generators correspond 4 to tetrads of points of Ω. Factoring this by a single short relation results in the Conway group ·O and the symmetric generators spontaneously reveal themselves to be the negatives of the Conway elements ξT.

    In the second case, we have n = 3795 and the symmetric generators correspond to the trios. The Lemma leads us to two simple relations and factoring by them results in the largest Janko group J4. In fact one of the relations is essentially redundant and John Bray has shown that omitting it leads to a group J4 × 2. In order to prove these assertions in the two cases, we first use known representations to verify that the group claimed is indeed an image of the progenitor and satisfies the additional relations. We then use the double coset enumerator of John Bray and the author, written in Magma, to show that the image has the right order. Unfortunately in both these cases the group N ≅ M24 is too small to enumerate double cosets of form NwN and so we use theoretical arguments to identify subgroups H ≅ 212 :  M24 and H ≅ 211 :  M24, respectively, and enumerate double cosets of form HwN.

  • NG2 cells generate oligodendrocytes and gray matter astrocytes in the spinal cord

  • Xiaoqin Zhu, Robert A. Hill, Akiko Nishiyama
  • Journal:
    Neuron Glia Biology / Volume 4 / Issue 1 / February 2008
    Published online by Cambridge University Press:
    13 November 2008, pp. 19-26
    Print publication:
    February 2008

  • NG2 cells represent a unique glial cell population that is distributed widely throughout the developing and adult CNS and is distinct from astrocytes, mature oligodendrocytes and microglia. The ability of NG2 cells to differentiate into myelinating oligodendrocytes has been documented in vivo and in vitro. We reported recently that NG2 cells in the forebrain differentiate into myelinating oligodendrocytes but into a subpopulation of protoplasmic astrocytes (Zhu et al., 2008). However, the in vivo fate of NG2 cells in the spinal cord and cerebellum has remained unknown. To investigate the fate of NG2 cells in caudal central nervous system (CNS) regions in vivo, we examined the phenotype of cells that express EGFP in mice that are double transgenic for NG2CreBAC and the Cre reporter Z/EG. The fate of NG2 cells can be studied in these mice by permanent expression of EGFP in cells that have undergone Cre-mediated recombination in NG2 cells. We find that NG2 cells give rise to oligodendrocytes in both gray and white matter of the spinal cord and cerebellum, and to protoplasmic astrocytes in the gray matter of the spinal cord. However, NG2 cells do not give rise to astrocytes in the white matter of the spinal cord and cerebellum. These observations indicate that NG2 cells serve as precursor cells for oligodendrocytes and a subpopulation of protoplasmic astrocytes throughout the rostrocaudal axis of the CNS.

  • Cellular proliferation and neurogenesis in the injured retina of adult zebrafish

  • DAVID A. CAMERON
  • Journal:
    Visual Neuroscience / Volume 17 / Issue 5 / September 2000
    Published online by Cambridge University Press:
    15 December 2000, pp. 789-797

  • The retinas of adult teleost fish can regenerate neurons following a chemical or mechanical injury. Previous studies have demonstrated that mechanical excision of fish retina induces a hyperplasia within the retinal sheet, including the formation of a proliferative blastema from whence new retinal cells are produced to fill the excision site. The current study was designed to address two issues regarding injury-induced retinal hyperplasia: (1) Retinas of adult zebrafish can regenerate following a surgical excision, but compared to other fish they contain very few proliferative cells: Might retinal injury in adult zebrafish therefore induce minimal, or perhaps no, hyperplasia? (2) The fate of injury-induced, proliferative retinal cells outside surgical excision sites has yet to be determined. Do such cells produce retinal neurons? Evidence is presented that mechanical injury to the adult zebrafish retina induces a dramatic increase in the number of proliferative cells both within and external to the lesion site, and some of these cells apparently migrate within the radial dimension of the retina. Evidence is also presented that injury-induced proliferative cells outside a lesion site can produce retinal neurons—including cone photoreceptors, interplexiform cells, and amacrine cells—that are incorporated into the extant retina. The results suggest that the adult zebrafish retina contains a latent population of cells that is induced to proliferate following retinal injury, and that these cells might represent a novel avenue for pluripotent neurogenesis within the intact adult teleost retina.