In this paper we formalized some theorems concerning the cyclic groups of prime power order. We formalize that every commutative cyclic group of prime power order is isomorphic to a direct product of family of cyclic groups [1], [18]. Let G be a finite group. The functor Ordset(G) yielding a subset of N is defined by the term (Def. 1) the set of all ord(a) where a is an element of G. One can ch...

In 1979, Rourke proposed to extend the set of cyclically reduced defining words of a group presentation P by using operations of cyclic permutation, inversion and taking double products. He proved that iterations of these operations yield all cyclically reduced words of the normal closure of defining words of P if the group, defined by the presentation P, is trivial. We generalize this result b...

We show that the epimorphism problem is solvable for targets are virtually cyclic or a product of an Abelian group and finite group.

The recently developed theory of Schur rings over a finite cyclic group is generalized to Schur rings over a ring R being a product of Galois rings of coprime characteristics. It is proved that if the characteristic of R is odd, then as in the cyclic group case any pure Schur ring over R is the tensor product of a pure cyclotomic ring and Schur rings of rank 2 over non-fields. Moreover, it is s...

To evaluate the protective effects of dexmedetomidine on colon and spinal cord rats with functional chronic visceral pain its regulatory mechanism extracellular signal-regulated kinase 1-cyclic adenosine monophosphate response element-binding protein signaling pathway. A rat model irritable bowel syndrome was established by colorectal distention stimulation. Animals were categorized into normal...

Faced with the need to find new drugs for all kinds of diseases, science sees that Nature offers numerous classes of compounds showing an impressively high biological potential. Among those are the cyclodepsipeptides, hybrid structures composed of amino and hydroxy acids. In the past decades numerous cyclodepsipeptides have been isolated and their potential as drugs has been studied extensively...

In this paper we study the Galois group of cover composition a q-cyclic étale and cyclic p-gonal for any odd prime p. Furthermore, give properties isogenous decompositions certain Prym Jacobian varieties associated to intermediate subcovers given by subgroups.

The propensity of oxoammonium cations to facilitate the oxidative ring-opening of cyclic ethers to their corresponding distal hydroxy ketones is investigated. The reaction has been evaluated using experimental and computational methods to gain deeper insight into trends in reactivity.