The abelian HSP Recall that in the hidden subgroup problem, we are given a function f : G→ S (for a known group G and a finite set S) satisfying f(x) = f(y) iff x and y are in the same (left) coset of the hidden subgroup H ≤ G. In this lecture we will use additive notation for the group operation of an abelian group, so we have f(x) = f(x) iff x − y ∈ H. The strategy for the general abelian HSP...

In this article, by using a partial on locally compact semi-direct product groups, we present a compatible extension of the Fourier transform. As a consequence, we extend the fundamental theorems of Abelian Fourier transform to non-Abelian case.

Tetris is a popular puzzle video game, invented in 1984. We formulate two versions of the game as transformation semigroup and use this formulation to view through lens Krohn-Rhodes theory. In variation upon which it restarts if player loses, we find permutation group structures, including symmetric \(S_5\) contains non-abelian simple subgroup. This implies, at least case, that iterated finitar...

In this paper, we show that for every abelian subgroup H of a Garside group, some conjugate gHg consists of ultra summit elements and the centralizer of H is a finite index subgroup of the normalizer of H. Combining with the results on translation numbers in Garside groups, we obtain an easy proof of the algebraic flat torus theorem for Garside groups and solve several algorithmic problems conc...

The undirected graphs atoms were introduced independently Mader[21] and Watkins [28] in order to show that the connectivity of a connected undirected vertex-transitive is large. The directed graphs atoms were introduced by Chaty in [3]. The author obtained a structure Theorem for the atoms in the last case [9], showing that the connectivity of a connected vertex-transitive directed is large. Th...

Let $pounds$ be the category of locally compact abelian groups and $A,Cin pounds$. In this paper, we define component extensions of $A$ by $C$ and show that the set of all component extensions of $A$ by $C$ forms a subgroup of $Ext(C,A)$ whenever $A$ is a connected group. We establish conditions under which the component extensions split and determine LCA groups which are component projective. ...

We prove that if F is a finitely generated abelian group of orientation preserving C1 diffeomorphisms of R2 which leaves invariant a compact set then there is a common fixed point for all elements of F . We also show that if F is any abelian subgroup of orientation preserving C1 diffeomorphisms of S2 then there is a common fixed point for all elements of a subgroup of F with index

We prove that if F is a finitely generated abelian group of orientation preserving C1 diffeomorphisms of R2 which leaves invariant a compact set then there is a common fixed point for all elements of F . We also show that if F is any abelian subgroup of orientation preserving C1 diffeomorphisms of S2 then there is a common fixed point for all elements of a subgroup of F with index

We shall consider diffeomorphism invariant theories within the Hamiltonian formulation, where space-time is assumed to be a topological product M ∼= Σ × R. The constraints of the theory will only generate the identity component D(Σ) of some subgroup D(Σ) of diffeomorphisms of Σ. This means that after a reduction by D(Σ), i.e., implementing the constraints – which itself is a highly non-trivial ...

We show that the commensurator of any quasiconvex abelian subgroup in a biautomatic group is small, sense it has finite image abstract subgroup. Using this criterion we exhibit groups are CAT(0) but not biautomatic. These also resolve number other questions concerning groups.