Browsing by Author "Rowley, Peter"

Now showing 1 - 5 of 5
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    2-Minimal subgroups in classical groups: Linear and unitary groups
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2011) Parker, Chris; Rowley, Peter
    [no abstract available]
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    A 3-Local characterization of Co2
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2008) Parker, Christopher; Rowley, Peter
    Conway's second largest simple group, Co2, is characterized by the centralizer of an element of order 3 and certain fusion data.
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    A 3-local identification of the alternating group of degree 8, the McLaughlin simple group and their automorphism groups
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2007) Parker, Christopher; Rowley, Peter
    In this article we give 3-local characterizations of the alternating and symmetric groups of degree 8 and use these characterizations to recognize the sporadic simple group discovered by McLaughlin from its 3-local subgroups.
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    Generating Finite Coxeter Groups with Elements of the Same Order
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2020) Hart, Sarah; Kelsey, Veronica; Rowley, Peter
    Supposing G is a group and k a natural number, dk(G) is defined to be the minimal number of elements of G of order k which generate G (setting dk(G)=0 if G has no such generating sets). This paper investigates dk(G) when G is a finite Coxeter group either of type Bn or Dn or of exceptional type. Together with Garzoni [3] and Yu [10], this determines dk(G) for all finite irreducible Coxeter groups G when 2≤k≤rank(G) (rank(G)+1 when G is of type An).
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    Root Cycles in Coxeter Groups
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2022) Hart, Sarah; Kelsey, Veronica; Rowley, Peter
    For an element w of a Coxeter group W there are two important attributes, namely its length, and its expression as a product of disjoint cycles in its action on Φ, the root system of W. This paper investigates the interaction between these two features of w, introducing the notion of the crossing number of w, κ(w). Writing w=c1⋯cr as a product of disjoint cycles we associate to each cycle ci a `crossing number' κ(ci), which is the number of positive roots α in ci for which w⋅α is negative. Let Seqk(w) be the sequence of κ(ci) written in increasing order, and let κ(w) = max Seqk(w). The length of w can be retrieved from this sequence, but Seqk(w) provides much more information. For a conjugacy class X of W let kmin(X)=min{κ(w)|w∈X} and let κ(W) be the maximum value of kmin across all conjugacy classes of W. We call κ(w) and κ(W), respectively, the crossing numbers of w and W. Here we determine the crossing numbers of all finite Coxeter groups and of all universal Coxeter groups. We also show, among other things, that for finite irreducible Coxeter groups if u and v are two elements of minimal length in the same conjugacy class X, then Seqk(u) = Seqk(v) and kmin(X)=κ(u)=κ(v). Also it is shown that the crossing number of an arbitrary Coxeter group is bounded below by the crossing number of a standard parabolic subgroup. Finally, examples are given to show that crossing numbers can be arbitrarily large for finite and infinite irreducible Coxeter groups.