We relate the connection between the sizes of circuits in suborbital graph for the normalizer of [Formula: see text] in PSL(2,[Formula: see text]) and the congruence equations arising from related group action. We give a number theoretic result which says that all prime divisors of [Formula: see text] for any integer u must be congruent to [Formula: see text].

There are errors in the proof of the uniqueness of arithmetic subgroups of the smallest covolume. In this note we correct the proof, obtain certain results which were stated as a conjecture and give several remarks on further development. 1.1. Let us recall some notations and basic notions. Following [B] we will assume that n is even. The group of orientation preserving isometries of the hyperb...

Let G be a semi-simple simply-connected complex algebraic group and T ⊂ B a maximal torus and a Borel subgroup respectively. Let h = Lie T be the Cartan subalgebra of the Lie algebra Lie G, and W := N(T )/T the Weyl group associated to the pair (G, T ), where N(T ) is the normalizer of T in G. We can view any element w = w mod T ∈ W as the element (denoted by the corresponding German character)...

Let p be a prime, k be a field of characteristic 6= p and N be the normalizer of the maximal torus in the projective linear group PGLn. We compute the exact value of the essential dimension edk(N ; p) of N at p for every n ≥ 1.

Let g be a complex simple Lie algebra. Fix a Borel subalgebra b and a Cartan subalgebra t ⊂ b. The nilpotent radical of b is denoted by u. The corresponding set of positive (resp. simple) roots is ∆ (resp. Π). An ideal of b is called ad-nilpotent, if it is contained in [b, b]. The theory of ad-nilpotent ideals has attracted much recent attention in the work of Kostant, Cellini-Papi, Sommers, an...

Herein are solutions to the exercises given in [1]. Let K be a field, and f(X) ∈ K[X] an irreducible polynomial. The root quantum number rK(f) is the number of roots of f in the stem field K(α), where α is an arbitrary choice of root of f . It was shown that rK(f) is well-defined and divides the degree of f . We start by establishing the connection with Galois theory. Let f be irreducible and s...

Bifurcation problems in which periodic boundary conditions (PBC) or Neumann boundary conditions (NBC) are imposed often involve partial differential equations that have Eu-clidean symmetry. In this case posing the bifurcation problem with either PBC or NBC on a finite domain can lead to a symmetric bifurcation problem for which the manifest symmetries of the domain do not completely characteriz...