Let L be a commutative Moufang loop (CML) with multiplication group M, and let F(L), F(M) be the Frattini subgroup and Frattini subgroup of L and M respectively. It is proved that F(L) = L if and only if F(M) = M and is described the structure of this CLM. Constructively it is defined the notion of normalizer for subloops in CML. Using this it is proved that if F(L) 6= L then L satisfies the no...

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].

The novel notion of rigid commutators is introduced to determine the sequence logarithms indices a certain normalizer chain in Sylow 2-subgroup symmetric group on 2^n letters. terms this are proved be those partial sums partitions an integer into at least two distinct parts, that relates famous Euler's partition theorem.

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...

this work is a continuation of [a‎. ‎o‎. ‎asar‎, ‎‎‎on infinitely generated groups whose proper subgroups are solvable‎, ‎{em j‎. ‎algebra}‎, ‎{bf 399} (2014) 870-886.]‎, ‎where it was shown‎ ‎that a perfect infinitely generated group whose proper subgroups‎ are solvable and in whose homomorphic images normal closures of ‎finitely generated subgroups are residually nilpotent is a fitting‎‎$p$-g...

WN ′ = W (N) N ′ denotes the corresponding Atkin–Lehner involution defined for Γ0(N). (If N ′ = 1, W1 means the identity operator.) Then we define the modular group Γ ∗ 0 (N) to be Γ ∗ 0 (N) = 〈Γ0(N) ∪ {WN ′}N ′ ‖N 〉, i.e., Γ ∗ 0 (N) is generated by Γ0(N) and {WN ′}N ′ ‖N . Then Γ ∗ 0 (N) is a normalizer of Γ0(N) in GL + 2 (Q) = {A ∈M2(Q) | detA > 0}. The factor group Γ ∗ 0 (N)/Γ0(N) is abelian...