Let G be a group and D be a subgroup of G. In this paper we are interested in the description of the lattice L(D; G) = fH j D H Gg of subgroups of G which contain D. We call subgroups H 2 L(D; G) subgroups intermediate between D and G. We are primarily interested in the parametrization of intermediate subgroups and their behaviour with respect to set-theoretic and group-theoretic operations lik...

We prove that there exists an integer p0 such that X split (p)(Q) is made of cusps and CM-points for any prime p > p0. Equivalently, for any non-CM elliptic curve E over Q and any prime p > p0 the image of the representation of Gal(¯ Q/Q) induced by the Galois action on the p-division points of E is not contained in the normalizer of a split Cartan subgroup. This gives a partial answer to an ol...

Given an infinite graph of profinite groups ( $${\cal G}$$ , Γ) we construct a $$\overline {\cal G} ,\overline \Gamma $$ ) such that Γ is densely embedded in the fundamental group $${\Pi _1}\left( {\overline } \right)$$ completion $${\pi {{\cal G},\Gamma and standard tree $$S\left( embeds . This answers Ribes’ question [5, Question 6.7.1]. Generalizing main results [8] [2] answer two other ques...

We prove that there exists an integer p0 such that X split (p)(Q) is made of cusps and CM-points for any prime p > p0. Equivalently, for any non-CM elliptic curve E over Q and any prime p > p0 the image of Gal(¯ Q/Q) by the representation induced by the Galois action on the p-division points of E is not contained in the normalizer of a split Cartan subgroup. This gives a partial answer to an ol...

We develop the theory of semisimplifications tensor categories defined by Barrett and Westbury. In particular, we compute semisimplification category representations a finite group in characteristic p terms normalizer its Sylow p-subgroup. This allows us to representation symmetric Sn+p p, where 0 ≤ n − 1, Deligne $$ \underline { \mathop {\mathrm {Rep}} \nolimits }^{\mathrm {ab}}S_t$$ , t ∈ℕ. a...

If G is a group, then subgroups A and B are commensurable if A ∩B has finite index in both A and B. The commensurator of A in G, denoted CommG(A), is {g ∈ G|(gAg−1) ∩A has finite index in both A and gAg−1}. It is straightforward to check that CommG(A) is a subgroup of G. A subgroup A is commensurated in G if CommG(A) = G. The centralizer of A in G is a subgroup of the normalizer of A in G which...

Normalizer circuits [1, 2] are generalized Clifford circuits that act on arbitrary finitedimensional systems Hd1 ⊗· · ·⊗Hdn with a standard basis labeled by the elements of a finite Abelian group G = Zd1 × · · · × Zdn . Normalizer gates implement operations associated with the group G and can be of three types: quantum Fourier transforms, group automorphism gates and quadratic phase gates. In t...

Normalizer circuits [3, 4] are a family of quantum circuits which generalize Cli ord circuits [5 8] to Hilbert spaces associated with arbitrary nite abelian groups G = Zd1 × · · · × Zdn . Normalizer circuits are composed of normalizer gates. Important examples are quantum Fourier transforms (QFTs), which play a central role in quantum algorithms, such as Shor's [9]. Refs. [3, 4] showed that nor...

In this study, we deal with the conjecture given in [R. Keskin, Suborbital graph for the normalizer of Γ0(m), European Journal of Combinatorics 27 (2006) 193206.], that when the normalizer of Γ0(N) acts transitively on Q ∪ {∞}, any circuit in the suborbital graph G(∞, u/n) for the normalizer of Γ0(N), is of the form v → T (v) → T (v) → · · · → T (v) → v, where n > 1, v ∈ Q ∪ {∞} and T is an ell...