In previous papers [4; 5; 6] we constructed and studied many classes of operations on groups which have the following properties: (1) Associative; (2) Commutative; (3) The resultant group contains isomorphic images of the original factors and is generated by them; (4) In the resultant group the intersection of a factor with normal subgroup generated by the remaining factors is the unit element....

In this paper among many other things we prove that the topological left amenability and left amenability of a weighted hypergroup (K, ?) are equivalent. For a normal subgroup H of K, we define a weight function ?? on KIH and obtain connection between left amenability of (K, ?) and (K|H, ??). Let H be a compact subhypergroup of K. We define the weight function on K||H and obtain connection...

Let H < PSL 2 ( Z ) be a finite index normal subgroup which is contained in principal congruence subgroup, and let Φ ≠ denote term of the lower central series or derived H. In this paper, we prove that commensurator R discrete. We thus obtain natural family thin subgroups whose commensurators are discrete, establishing some cases conjecture Shalom.

A nilpotent group G is a finite group that is the direct product of its Sylow p-subgroups. Theorem 1.1 (Fitting's Theorem) Let G be a finite group, and let H and K be two nilpotent normal subgroups of G. Then HK is nilpotent. Hence in any finite group there is a unique maximal normal nilpotent subgroup, and every nilpotent normal subgroup lies inside this; it is called the Fitting subgroup, and...

A brief comment about items (3) and (4). If H is a subgroup of G the inclusion map is an injective homomorphism from H to G . On the other hand, the image of an injective homomorphism α : H −→ G is a subgroup that is isomorphic to H . So the study of injective homomorphisms is (up to isomorphism) the study of subgroups. After studying subgroups, we defined normal subgroup and showed several equ...

We provide a new and simple proof to show that finite group in which every non-nilpotent maximal subgroup is normal solvable.

In this paper we introduce a new definition of the first non-abelian cohomology of topological groups.  We relate the cohomology of a normal subgroup $N$ of a topological group $G$ and the quotient $G/N$ to the cohomology of $G$. We get the inflation-restriction exact sequence. Also, we obtain a seven-term exact cohomology sequence up to dimension 2. We give an interpretation of the first non-a...

Provisional hypomyopathic juvenile dermatomyositis is a subgroup of clinically amyopathic juvenile dermatomyositis provisional. The diagnostic criteria include: Classic dermatomyositis skin lesions – which have to be confirmed by biopsy –, no involvement of proximal muscles, subclinical involvement of these muscles and normal level of muscle enzymes. These criteria should be present for a minim...