Trivial Group Research Articles

The green correspondents of simple group modules

We shall show that these correspondents are not arbitrary modules, but have a strong property with respect to specializations (see Corollary 2.8) which ensures that they belong to a family of modules that are theoretically classifiable using only local information about the group (see Corollary 3.12 and the remarks following it). This extends to arbitrary simple modular group modules the similar properties proven in [2] for such modules lying in blocks with trivial intersection defect groups. Indeed, we lean heavily on [2], and follow its notation as much as possible. As an addendum we indicate in Section 4 how our results can be extended from Green correspondents to sources of simple group modules.

Probability measures with trivial Stam groups

Let μ be a probability measure on the real line ℝ, x a real number and δ(x) the probability atom concentrated at x. Stam made the interesting observation that eitheror else(ii) δ(x)* μn, are mutually singular for all positive integers n.

Integral representations with trivial first cohomology groups

Let Π be a finite group and denote by MΠ the class of finitely generated Z-free ZΠ-modules. In [2] we defined a certain equivalence relation on MΠ and constructed the abelian semigroup T(Π), which was studied in [3] (see [1] and [5], too).

On ternary self-dual codes of length 24

A partial classification is given of the self-dual codes of length 24 over GF (3). The main results are as follows: there are exactly two codes with minimum Hamming distance d=9 ; most of the codes have d=6 and are indecomposable; one code with d=6 has a trivial automorphism group (this is the first such self-dual code that has been found); the codes generated by the 59 inequivalent 24 \times 24 Hadamard matrices have been investigated and there appear to be only nine inequivalent codes (two with d=9 and seven with d=6) ; and in all there are 27 decomposable codes, at least 96 indecomposable codes with d=6 , and the total number of inequivalent codes is at least 140 .

The class group of a one-dimensional affinoid space

A curve X over a non-archimedean valued field is with respect to its analytic structure a finite union of affinoid spaces. The main result states that the class group of a one dimensional, connected, regular affinoid space Y is trivial if and only if Y is a subspace of P 1 . As a consequence, X has locally a trivial class group if and only if the stable reduction of X has only rational components.

Trivial intersection groups

Trivial intersection groups

Finite groups with trivial class groups

Finite groups with trivial class groups

P-DIVISIBLE GROUPS OVERZ

In this paper we construct nontrivial 2-divisible groups over Z which are isogenous to trivial groups and prove the following: THEOREM. If the height h of a 2-divisible group {G(v)} over Z is at most 4, then {G(v)} is isogenous to a trivial group.

An Algorithm for the Vertex Packing Problem

This paper presents several properties of the vertex packing problem and develops an algorithm for its solution. We show a relationship between the vertex packing problem on the original graph and on a class of related bipartite graphs. This can be used to generate good initial solutions, which in some cases are known to be optimal. We also show that the group-theoretic approach to integer programming, when applied to vertex packing, yields a trivial group problem regardless of the determinant of the associated basis matrix. These ideas are incorporated into an algorithm and computational experience is presented.

Functionally complete groups

A group G is called functionally complete if for an arbitrary natural number n every mapping f: Gn → G can be realized by a “polynomial” in at most n variables over the group G. We know that a group G is functionally complete if and only if it is either trivial or a finite simple non-Abelian group [Ref. Zh. Mat. 9A174 (1975)]. In this article the “degree” of a polynomial and the connected notions of n-functional completeness, (n; k1, ..., kn)-functional completeness, and strong functional completeness are introduced. It is shown that for n > 1 these notions and the notion of functional completeness are equivalent, and apart from all finite simple non-Abelian groups, only the trivial group and groups of second order are 1-functionally complete.

Spin point groups

The 598 classes of nontrivial spin point groups are derived and tabulated. The relationship between classes of nontrivial spin point groups and the 32 classes of trivial magnetic point group and the 58 classes of nontrivial magnetic point groups is also given.

Phase III crystal structure and 115 K phase transition of hexamethylbenzene

High-resolution spectroscopic data-Raman, infrared, and electronic-have been employed for the study of low-temperature, phase III, hexamethylbenzene and its associated X-type phase transition. Phase III is trigonal, with point group $6 and one molecule per primitive unit cell, giving an $6 site group, an $6 molecular group, and a trivial interchange group. The X-type phase transition is not second order, and it is not related to a free rotation of the methyl groups. Restricted, geared methyl-group rotations do exist both above and below the phase transition temperature, and are sensitive to it. The molecular point symmetry is $6 in phase III and effectively so in phase II.

On group actions on groups and associated series

1. Introduction. In this paper we consider Q-groups; that is, Q is a group and we consider groups N endowed with a Q-action, meaning a homomorphism of Q into the group of automorphisms of N. In (4) a lower central Q-series was defined for such a Q-group, N, generalizing the lower central series of a group, and results were obtained relating to the localization of such a series. Since the ideas in that paper were inspired by the homotopical localization theory of nilpotent spaces (see (6)), the main body of results in (4) was concerned with the case in which N is nilpotent, and perhaps also the group Q and the action of Q on N (in the sense that the lower central Q-series terminates after a finite number of steps with the trivial group {1}). We now adopt a broader view-point and only restrict ourselves to the nilpotent case when our results appear to require us to do so; thus the spirit of this paper is much more that of general group theory as presented in [(8), especially ch. VI]. Thus, while there is some overlap of results, the methods used are not the same and many results (for example, Theorem 3·1) are far more general than any obtained in (4). Moreover, the methods also appear to us to be more appropriate in that essential appeal was made in (4) to a sophisticated theorem of Norman Blackburn on nilpotent groups, whereas here we merely use homological methods, the construction of the semidirect product and, in section 4, some very classical facts of the commutator calculus.

2-Divisible groups over Z

In this paper we construct nontrivial 2-divisible groups over Z which are isogenous to trivial groups and prove the following: THEOREM. If the height h of a 2-divisible group {G(v)} over Z is at most 4, then {G(v)} is isogenous to a trivial group.

On the Existence of Totally Inhomogeneous Spaces

A property of total inhomogeneity for topological spaces is defined and is shown to be stronger than that of rigidity (i.e., of having trivial autohomeomorphism group). It is further shown that compact, rigid, Hausdorff spaces are totally inhomogeneous, and that totally inhomogeneous spaces exist in profusion as dense subspaces of a class of locally compact spaces.

On the existence of totally inhomogeneous spaces

A property of total inhomogeneity for topological spaces is defined and is shown to be stronger than that of rigidity (i.e., of having trivial autohomeomorphism group). It is further shown that compact, rigid, Hausdorff spaces are totally inhomogeneous, and that totally inhomogeneous spaces exist in profusion as dense subspaces of a class of locally compact spaces.

Homotopy groups of compact Hausdorff spaces with trivial shape

Given a collection { π n : n = 1 , 2 , ⋯ } \{ {\pi _n}:n = 1,2, \cdots \} of countable groups such that π i {\pi _i} is abelian and admits π 1 {\pi _1} as a group of operators for i ≧ 2 i \geqq 2 , we construct here an arcwise connected compact metric space of trivial shape whose j j th homotopy group is isomorphic to π j {\pi _j} for j = 1 , 2 , ⋯ j = 1,2, \cdots . The isomorphisms preserve the action of the first group on the higher groups. Thus, the homotopy groups of a compact metric space of trivial shape may be quite arbitrary.

A class of geometric lattices based on finite groups

For any finite group G and positive integer n a finite geometric lattice Q n (G) of rank n , the lattice of partial G -partitions, is constructed. Let P n+1 be the lattice of partitions of an ( n +1)-set. There exists a surjection π: Q n (G) → P n+1 , and an injection t: P n+1 → Q n+1 (G) , each of which preserves order and rank. When G is the trivial group, π=t −1 reduces to an isomorphism. The interval structure, Möbius function, and characteristic polynomial of Q n (G) are determined, and Stirling-like identities for the Whitney numbers obtained. The existence of a Boolean sublattice of modular elements in Q n (G) is established, implying that Q n (G) is supersolvable. It is further shown that non-isomorphic groups give non-isomorphic lattices, and the representation problem is solved completely: Q n (G) is representable over a field when n ≥3 if and only if G is isomorphic to a subgroup of the multiplicative group of the field. Consequently, Q n (G) is representable over no field iff G is noncyclic, and, if G is cyclic of order m , then Q n (G) is representable over (a) every field iff m =1, (b) a finite field of order q iff m divides q −1, (c) the rational or real fields iff m =1 or 2, and (d) the complex field for all m .

On 2-sylow intersections

Letz be an involution in the finite groupG and suppose thatz belongs to the center of a Sylow subgroup ofG. Ifz belongs to a unique Sylow subgroup ofG and ifG is not a trivial intersection group, thenG is not a simple group.

On Inflation and Torsion of Amitsur Cohomology

Studies of torsion [2] and inflation [11] of Amitsur cohomology have primarily been concerned with module-finite faithful projective algebras. In this paper, our goal is to consider these topics for more general algebras. The fundamental tool, in case R is a domain with quotient field K, is the functor UK/U (defined in § 1), together with the monomorphic connecting map of Amitsur cohomology H1(S/R, UK/U) → H2(S/R, U) arising from a (commutative) flat R-algebra S and the unit functor U. This map was first considered in [10, Chapter IV, Theorem 1.6], where it was proved to be an isomorphism for certain étale faithfully flat algebras S in case R is an algebraic number ring with trivial Brauer group. In Corollary 1.5 it is shown, in the case of a module-finite faithful projective S over a regular domain R, that the connecting map is the kernel of the canonical homomorphism [8] from H2(S/R, U) to the split Brauer group B(S/R). As the latter map is often injective [8, Corollary 7.7], one might expect instances of vanishing of H1(S/R, UK/U) without assuming both R Noetherian and S module-finite R-projective. Much of § 1 (viz. (1.7)-(1.9)) is devoted to such examples.