• Arithmetic results on orbits of linear groups

      Giudici, Michael; Liebeck, Martin W.; Praeger, Cheryl E.; Saxl, Jan; Tiep, Pham Huu; Univ Arizona, Dept Math (AMER MATHEMATICAL SOC, 2015-08-19)
      Let p be a prime and G a subgroup of GL(d)(p). We define G to be p-exceptional if it has order divisible by p, but all its orbits on vectors have size coprime to p. We obtain a classification of p-exceptional linear groups. This has consequences for a well-known conjecture in representation theory, and also for a longstanding question concerning 1/2-transitive linear groups (i.e. those having all orbits on nonzero vectors of equal length), classifying those of order divisible by p.
    • Finite groups with odd Sylow normalizers

      Guralnick, Robert M.; Navarro, Gabriel; Tiep, Pham Huu; Univ Arizona, Dept Math (AMER MATHEMATICAL SOC, 2016-06-10)
      We determine the non-abelian composition factors of the finite groups with Sylow normalizers of odd order. As a consequence, among others, we prove the McKay conjecture and the Alperin weight conjecture for these groups at these primes.
    • Nilpotent and abelian Hall subgroups in finite groups

      Beltrán, Antonio; Felipe, María José; Malle, Gunter; Moretó, Alexander; Navarro, Gabriel; Sanus, Lucia; Solomon, Ronald; Tiep, Pham Huu; Univ Arizona, Dept Math (AMER MATHEMATICAL SOC, 2015-07-10)
      We give a characterization of the finite groups having nilpotent or abelian Hall pi-subgroups that can easily be verified using the character table.

      Gillette, Andrew; Kloefkorn, Tyler; Univ Arizona, Dept Math (AMER MATHEMATICAL SOC, 2019-03)
      We introduce the family of trimmed serendipity finite element differential form spaces, defined on cubical meshes in any number of dimensions, for any polynomial degree, and for any form order. The relation between the trimmed serendipity family and the (non-trimmed) serendipity family developed by Arnold and Awanou [Math. Comp. 83 (2014), pp. 1551-1570] is analogous to the relation between the trimmed and (non-trimmed) polynomial finite element differential form families on simplicial meshes from finite element exterior calculus. We provide degrees of freedom in the general setting and prove that they are unisolvent for the trimmed serendipity spaces. The sequence of trimmed serendipity spaces with a fixed polynomial order r provides an explicit example of a system described by Christiansen and Gillette [ESAIM: M2AN 50 (2016), pp. 883-850], namely, a minimal compatible finite element system on squares or cubes containing order r - 1 polynomial differential forms.