One-parameter subgroups of the generalized Poincaré group p(2, n) and their invariants

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 9, pp. 1169–1172, September, 1989.     

Finite groups acting 2-transitively on their sylow subgroups

In this paper we have completely determined: (1) all almost simple groups which act 2-transitively on one of their sets of Sylow p-subgroups. (2) all non-abelian simple groups T whose automorphism group acts 2-transitively on one of the sets of Sylow p-subgroups of T. (3) all finite groups which are 2-transitive on all their sets of Sylow subgroups. The first author acknowledges the support of OPR Scholarship of Australia The second author is supported by the National Natural Science Foundation of China. Thanks are also due to the Department of Mathematics, the University of Western Australia, where he did his part of this work for its hospitality     

The covering radius of PGL2(q)

The covering radius of a subset $C$ of the symmetric group ${\mathrm{S}}_n$ is the maximal Hamming distance of an element of ${\mathrm{S}}_n$ from $C$. This note determines the covering radii of the finite $2$-dimensional projective general linear groups. It turns out that the covering radius of ${\mathrm{PGL}}_2\left(q\right)$ is $q?2$ if $q$ is even, and is $q?3$ if $q$ is odd.     

Relation of orbital integrals on SO(5) and PGL(2)

We relate the “Fourier” orbital integrals of corresponding spherical functions on thep-adic groups SO(5) and PGL(2). The correspondence is defined by a “lifting” of representations of these groups. This is a local “fundamental lemma” needed to compare the geometric sides of the global Fourier summation formulae (or relative trace formulae) on these two groups. This comparison leads to conclusions about a well known lifting of representations from PGL(2) to PGSp(4). This lifting produces counter examples to the Ramanujan conjecture.     

Irreducible subgroups of SL(n,K) generated by root subgroups

Jack McLaughlin determines all the irreducible subgroups of SL(n, K) generated by root subgroups for commutative fields K. In this paper I do the same thing for noncommutative division rings K.     

Torus period coefficients on PGL(2) and Dirichlet series

We fix a quadratic number field E, and the corresponding torus in $\mathrm{PGL}(2)$. We consider twisted (by a Hecke character of the field E) torus periods of automorphic functions. We prove meromorphic continuation for a Dirichlet series generated by these twisted periods.     

Tetravalent Edge-transitive Cayley Graphs of PGL （2, p）

t Let F = Cay（G, S）, R（G） be the right regular representation of G. The graph Г is called normal with respect to G, if R（G） is normal in the full automorphism group Aut（F） of F. Г is called a bi-normal with respect to G if R（G） is not normal in Aut（Г）, but R（G） contains a subgroup of index 2 which is normal in Aut（F）. In this paper, we prove that connected tetravalent edge-transitive Cayley graphs on PGL（2,p） are either normal or bi-normal when p ≠ 11 is a prime.     

The Antipode and the (co)invariants of a Finite Hopf (co)quasigroup

If $\mathcal{C}$ is a strict braided monoidal category in which the idempotent morphisms have a canonical split, we find conditions under which the antipode of a Hopf (co)quasigroup is an isomorphism. We also prove that the object of (co)invariants of a finite Hopf (co)quasigroup in $\mathcal{C}$ is an invertible object.     

Finite type invariants and fatgraphs

We define an invariant ∇G(M) of pairs M,G, where M is a 3-manifold obtained by surgery on some framed link in the cylinder Σ×I, Σ is a connected surface with at least one boundary component, and G is a fatgraph spine of Σ. In effect, ∇G is the composition with the ιn maps of Le-Murakami-Ohtsuki of the link invariant of Andersen-Mattes-Reshetikhin computed relative to choices determined by the fatgraph G; this provides a basic connection between 2d geometry and 3d quantum topology. For each fixed G, this invariant is shown to be universal for homology cylinders, i.e., ∇G establishes an isomorphism from an appropriate vector space of homology cylinders to a certain algebra of Jacobi diagrams. Via composition for any pair of fatgraph spines G,G^′ of Σ, we derive a representation of the Ptolemy groupoid, i.e., the combinatorial model for the fundamental path groupoid of Teichmüller space, as a group of automorphisms of this algebra. The space comes equipped with a geometrically natural product induced by stacking cylinders on top of one another and furthermore supports related operations which arise by gluing a homology handlebody to one end of a cylinder or to another homology handlebody. We compute how ∇G interacts with all three operations explicitly in terms of natural products on Jacobi diagrams and certain diagrammatic constants. Our main result gives an explicit extension of the LMO invariant of 3-manifolds to the Ptolemy groupoid in terms of these operations, and this groupoid extension nearly fits the paradigm of a TQFT. We finally re-derive the Morita-Penner cocycle representing the first Johnson homomorphism using a variant/generalization of ∇G.     

Dense subgroups ofC(K)-Stone-Weierstrass-type theorems for groups

LetB be a subgroup ofC(K) which separates points and contains the constants. An elementh∈C(R) operates onB iff∈B implies thath°f∈B. An elementh∈C(R) is condensing if its operation onB implies the density ofB inC(K). Similar notation applies to subgroupsB ofC(K, G) whereG is a metrizable group. We study the setD(G) of condensing functions inC(G, G) whenG is the additive group of a real Banach space and in particular whenG=R ⁿ.