We study the group of automorphisms of Shimura curves X0(D,N) attached to an Eichler order of square-free level N in an indefinite rational quaternion algebra of discriminant D > 1. We prove that, when the genus g of the curve is greater than or equal to 2, Aut(X0(D,N)) is a 2-elementary abelian group which contains the group of Atkin-Lehner involutions W0(D,N) as a subgroup of index 1 or 2. It...

The holomorph of a group G is NormB(λ(G)), the normalizer of the left regular representation λ(G) in its group of permutations B = Perm(G). The multiple holomorph of G is the normalizer of the holomorph in B. The multiple holomorph and its quotient by the holomorph encodes a great deal of information about the holomorph itself and about the group λ(G) and its conjugates within the holomorph. We...

In this thesis, we introduce new models of quantum computation to study the potential and limitations of quantum computer algorithms. Our models are based on algebraic extensions of the qubit Clifford gates (CNOT, Hadamard and π/4-phase gates) and Gottesman’s stabilizer formalism of quantum codes. We give two main kinds of technical contributions with applications in quantum algorithm design, c...

We determine the group structure of the normalizer of Γ0(N) in SL2(R) modulo Γ0(N). These results correct the Atkin-Lehner statement [1, Theorem 8].

let $g$ be a finite group‎. ‎a subgroup‎ ‎$h$ of $g$ is called an $mathcal h $ -subgroup in‎ ‎$g$ if $n_g (h)cap h^gleq h$ for all $gin‎ ‎g$. a subgroup $h$ of $g$ is called a weakly‎ $mathcal h^ast $-subgroup in $g$ if there exists a‎ ‎subgroup $k$ of $g$ such that $g=hk$ and $hcap‎ ‎k$ is an $mathcal h$-subgroup in $g$. we‎ ‎investigate the structure of the finite group $g$ under the‎ ‎assump...

Alperin [I] has recently introduced a fundamental method for conjugating from one p-Sylow subgroup Q of a finite group G to a second p-Sylow subgroup Pin a series of steps. The importance of this is that it provides information concerning the conjugacy in G of subsets of P, that is, fusion of subsets of P. In certain situations it is possible to extend these results to the case of conjugate p-s...

Most interactive theorem provers based on Type Theory automatically check termination of function definitions, and thus restrict to structurally terminating ones. It follows that implementing a full normalizer for the λ-calculus in an interactive theorem prover, and thus establishing formal properties on it, is a difficult issue. Relying on hereditary substitutions, that are structurally termin...

A X g. We exhibit natural embeddings of M(A X) in M(A) that give rise to monomorphisms from 1 (M(A X)) to 1 (M(A)). We call the images of these monomorphisms intersection subgroups of type X and prove that they form a conjugacy class of subgroups of 1 (M(A)). Recall that X in L(A) is modular if X+Y is an element of L(A) for all Y in L(A). We call X in L(A) supersolvable if there exists a chain ...