Normal Subgroups Of Groups Research Articles

Quasi-subtractive varieties

AbstractVarieties like groups, rings, or Boolean algebras have the property that, in any of their members, the lattice of congruences is isomorphic to a lattice of more manageable objects, for example normal subgroups of groups, two-sided ideals of rings, filters (or ideals) of Boolean algebras. Abstract algebraic logic can explain these phenomena at a rather satisfactory level of generality: in every memberAof aτ-regular varietythe lattice of congruences ofAis isomorphic to the lattice of deductive filters onAof theτ-assertional logic of. Moreover, ifhas a constant 1 in its type and is 1-subtractive, the deductive filters onA∈of the 1-assertional logic ofcoincide with the-ideals ofAin the sense of Gumm and Ursini, for which we have a manageable concept of ideal generation.However, there are isomorphism theorems, for example, in the theories of residuated lattices, pseudointerior algebras and quasi-MV algebras that cannot be subsumed by these general results. The aim of the present paper is to appropriately generalise the concepts of subtractivity andτ-regularity in such a way as to shed some light on the deep reason behind such theorems. The tools and concepts we develop hereby provide a common umbrella for the algebraic investigation of several families of logics, including substructural logics, modal logics, quantum logics, and logics of constructive mathematics.

ON THE EXISTENCE OF NON-ABELIAN (210, 77, 28), (336, 135, 54) AND (496, 55, 6) DIFFERENCE SETS

Lander [Symmetric Design: An Algebraic Approach, London Mathematical Society Lecture Note Series, Vol. 74 (Cambridge University Press, Cambridge, 1983)] showed that some abelian (v, k, λ) difference sets do not exist in some groups of order v and listed some parameters of difference sets that were open for small values of k (k ≤ 50). Various authors have since studied the open cases and have either constructed the difference sets when they exist or proved their non-existence. Using restrictions imposed by the underlying normal subgroups of groups, algebraic number theory, group representations and rational idempotents of the group ring, we rule out the existence of (496, 55, 6) difference sets and show that non-abelian (210, 77, 28) and (336, 135, 54) difference sets do not exist in most groups of 210 and 336, respectively.

Normal subgroups of diffeomorphism and homeomorphism groups of ℝn and other open manifolds

AbstractWe determine all the normal subgroups of the group of Cr diffeomorphisms of ℝn, 1≤r≤∞, except when r=n+1 or n=4, and also of the group of homeomorphisms of ℝn ( r=0). We also study the group A0 of diffeomorphisms of an open manifold M that are isotopic to the identity. If M is the interior of a compact manifold with non-empty boundary, then the quotient of A0 by the normal subgroup of diffeomorphisms that coincide with the identity near to a given end e of M is simple.

The divisible radical of a group

Abstract We consider the existence or otherwise of canonical divisible normal subgroups of groups in general. We present more counterexamples than positive results. These counterexamples constitute the substantive part of this paper.

Normal Subgroups of Groups Which Split Over The Infinite Cyclic Group

Let G be either a free product with amalgamation A*CB or an HNN group A*C, where all normal subgroups of C are finitely generated. Suppose that both A and B have no non-trivial finitely generated normal subgroups of infinite indices. We show that if G contains a finitely generated normal subgroup N which intersects A or B non-trivially but is not contained in C, then the index of N in G is finite. 1991 Mathematics Subject Classification 20E06.

Towards a system in space group representations

The structure of representations of a group is tightly connected with the lattice of normal subgroups of that group. We show at first how the correlation between irreducible representations and their kernels can be used if we know the position of kernels in the lattice of normal subgroups. The rest of paper concerns the problems of deciphering the lattices of normal subgroups of space and subperiodic groups. We show that these lattices can be described easily for irreducible space groups. In case of reducible space groups we show how to get a partial description of the lattice with use of sublattices which are isomorphic images of lattices of subperiodic groups. Finally we show the ways for derivation and description of lattices of normal subgroups of subperiodic groups; in our case of layer and rod groups. Representative examples of lattices illustrate particular steps of the development of the whole scheme.

Normal Subgroups of Groups which are Products of Two Abelian Subgroups

It is shown that if a group $G = AB$, where $A$ and $B$ are Abelian subgroups of $G,A \ne B$, and either $A$ or $B$ satisfies the maximum condition, then there is a normal subgroup $N$ of $G$, $N \ne G$, such that $N$ contains either $A$ or $B$.

Normal subgroups of groups which are products of two Abelian subgroups

It is shown that if a group G = A B G = AB , where A A and B B are Abelian subgroups of G , A ≠ B G,A \ne B , and either A A or B B satisfies the maximum condition, then there is a normal subgroup N N of G G , N ≠ G N \ne G , such that N N contains either A A or B B .