We provide examples of groups which are indecomposable by direct product, and more generally which are uniquely decomposable as direct products of indecomposable groups. Examples include Coxeter groups, for which we give an alternative approach to recent results of L. Paris. For a finitely generated linear group Γ, we establish an upper bound on the number of factors of which Γ can be the direc...

Pseudo-Galois extensions are shown to be depth two extensions. Studying its left bialgebroid, we construct an enveloping Hopf algebroid for the semi-direct product of groups or involutive Hopf algebras and their module algebras. It is a type of cofibered sum of two inclusions of the Hopf algebra into the semi-direct product and its derived right crossed product. Van Oystaeyen and Panaite observ...

We obtain a kind of structure theorem for the automorphism group AutA of a unital C-algebra A. According to it, AutA can be regarded as a subgroup of the semi-direct product of direct product group consisting of some family of projective unitary groups and some permutation group on the spectrum of A.

In this paper, we consider the hidden subgroup problem (HSP) over the class of semi-direct product groups Zn⋊Zq. The definition of the semi-direct product depending on the choice of an homomorphism, we first analyze the different possibilities for this homomorphism in function of n and q. Then, we present a polynomial-time quantum algorithm for the case Zpr ⋊ Zp when p is an odd prime.

In this paper, we consider the hidden subgroup problem (HSP) over the class of semi-direct product groups Zpr ⋊ Zq, for p and q prime. We first present a classification of these groups in five classes. Then, we describe a polynomial-time quantum algorithm solving the HSP over all the groups of one of these classes: the groups of the form Zpr ⋊ Zp, where p is an odd prime. Our algorithm works ev...

In this paper, we consider the hidden subgroup problem (HSP) over the class of semi-direct product groups Zn ⋊ Zq. The definition of the semi-direct product depending on the choice of an homomorphism, we first analyze the different possibilities for this homomorphism in function of n and q. Then, we present a polynomial-time quantum algorithm solving the HSP over the groups of the form Zpr ⋊ Zp...