Noether’s problem then asks whether K(G) is rational (= purely transcendental) over K. Noether’s problem is closely related to the inverse Galois problem. The main results about Noether’s problem for abelian groups can be found in the survey article [Sw]. More recently, Noether’s problem for nonabelian p-groups was investigated in [CHK, CHPK, HK2, Ka1, Ka2]. Let n ≥ 2 be an arbitrary natural nu...

Let $A$ be an abelian topological group and $B$ a trivial topological $A$-module. In this paper we define the second bilinear cohomology with a trivial coefficient. We show that every abelian group can be embedded in a central extension of abelian groups with bilinear cocycle. Also we show that in the category of locally compact abelian groups a central extension with a continuous section can b...

The triple factorization of a group $G$ has been studied recently showing that $G=ABA$ for some proper subgroups $A$ and $B$ of $G$, the definition of rank-two geometry and rank-two coset geometry which is closely related to the triple factorization was defined and calculated for abelian groups. In this paper we study two infinite classes of non-abelian finite groups $D_{2n}$ and $PSL(2,2^{n})$...

‎suppose $n$ is a fixed positive integer‎. ‎we introduce the relative n-th non-commuting graph $gamma^{n} _{h,g}$‎, ‎associated to the non-abelian subgroup $h$ of group $g$‎. ‎the vertex set is $gsetminus c^n_{h,g}$ in which $c^n_{h,g} = {xin g‎ : ‎[x,y^{n}]=1 mbox{~and~} [x^{n},y]=1mbox{~for~all~} yin h}$‎. ‎moreover‎, ‎${x,y}$ is an edge if $x$ or $y$ belong to $h$ and $xy^{n}eq y^{n}x$ or $x...

We examine the existence of universal elements in classes of innnite abelian groups. The method is deening some invariant of a group relative to a club guessing sequence, a combinatorial tool marketed here to algebraists. We prove, for example: Theorem: For n 2, there is a purely universal separable p-group in @ n if, and only if, 2 @ 0 @ n. + < < @ 0 , then there is no universal separable p-gr...

Let E be an elliptic curve over a field K with a discrete valuation v with residue class field k. Suppose E has 'additive reduction' at o, i.e. the connected component A ° of the special fibre A0 of the N6ron minimal model is isomorphic to Q . Then the order of Ao(k)/A°(k) is at most 4 as can be seen by inspection of the usual tables, cf. [9, pp. 124-125] and [5, p. 46]. Thus it follows that if...