Vanishing-off subgroups and supercharacter theory products

The vanishing-off subgroup

In this paper, we define the vanishing-off subgroup of a nonabelian group. We study the structure of the quotient of this subgroup and a central series obtained from this subgroup. MSC primary: 20C15, MSC secondary: 20D25

Maximal subgroups of direct products

A group G is simple if and only if the diagonal subgroup of G ×G is a maximal subgroup. This striking property is very easy to prove and raises the question of determining all the maximal subgroups of G , where G denotes the direct product of n copies of G . The first purpose of this note is to answer completely this question. We show in particular that if G is perfect, then any maximal subgrou...

Two subgroups and semi-direct products

Proof. Clearly, the image of F is HK by definition. To see that F is injective, suppose that F (h1, k1) = F (h2, k2). Then by definition h1k1 = h2k2. Thus h −1 2 h1 = k2k −1 1 . Since H is a subgroup, h−1 2 h1 ∈ H, and since K is a subgroup, k2k −1 1 ∈ K. Thus h−1 2 h1 = k2k −1 1 ∈ H ∩ K = {1}, and so h −1 2 h1 = k2k −1 1 = 1. It follows that h−1 2 h1 = 1, so that h1 = h2, and similarly k2k −1 ...

Non - vanishing of scalar products of Fourier -

A vanishing theorem for the homology of discrete subgroups

For any discrete, torsion-free subgroup Γ of Sp(n, 1) (resp. F−20 4 ) with no parabolic elements, we prove that H4n−1(Γ;V ) = 0 (resp. Hi(Γ;V ) = 0 for i = 13, 14, 15) for any Γ–module V . The main technical advance is a new bound on the p–Jacobian of the barycenter map of Besson–Courtois–Gallot. We also apply this estimate to obtain an inequality between the critical exponent and homological d...