Extensions Of Abelian Groups Research Articles

Hypercentral extensions of Abelian groups

Hypercentral extensions of Abelian groups

Immediate extensions of abelian groups

Immediate extensions of abelian groups

The occurrence problem for extensions of Abelian groups by nilpotent groups

The occurrence problem for extensions of Abelian groups by nilpotent groups

A Lemma on Extensions of Abelian Groups

A Lemma on Extensions of Abelian Groups

Abelian group extensions and the axiom of constructibility

Abelian group extensions and the axiom of constructibility

Notes on Minimality and Ergodicity of Compact Abelian Group Extensions of Dynamics

Notes on Minimality and Ergodicity of Compact Abelian Group Extensions of Dynamics

Extensions of Abelian Groups of Finite Rank

Every abelian group $X$ of finite rank arises as the middle group of an extension $e:0 \to F \to X \to T \to 0$ where $F$ is free of finite rank $n$ and $T$ is torsion with the $p$-ranks of $T$ finite for all primes $p$. Given such a $T$ and $F$ we study the equivalence classes of such extensions which result from stipulating that two extensions ${e_i}:0 \to F \to {X_i} \to T \to 0,i = 1,2$, are equivalent if ${e_1} = \beta {e_2}\alpha$ for $\alpha \in \operatorname {Aut} (T)$ and $\beta \in \operatorname {Aut} (F)$. We reduce the problem to $T$ $p$-primary of finite rank, where in the one case $T$ is injective, and in the other case $T$ is reduced. Suppose $T = \Pi _{i = 1}^m{T_i}$. In our main theorems we prove that in each case these equivalence classes of extensions are in 1-1 correspondence with the equivalence classes of $n$-generated subgroups of $E$ where $E = \Pi _{i = 1}^m{E_i},{E_i} = \operatorname {End} ({T_i})$. Two $n$-generated subgroups of $E$ will be called equivalent if one can be mapped onto the other by an automorphism of $E$.

Extensions of abelian groups of finite rank

Every abelian group $X$ of finite rank arises as the middle group of an extension $e:0 \to F \to X \to T \to 0$ where $F$ is free of finite rank $n$ and $T$ is torsion with the $p$-ranks of $T$ finite for all primes $p$. Given such a $T$ and $F$ we study the equivalence classes of such extensions which result from stipulating that two extensions ${e_i}:0 \to F \to {X_i} \to T \to 0,i = 1,2$, are equivalent if ${e_1} = \beta {e_2}\alpha$ for $\alpha \in \operatorname {Aut} (T)$ and $\beta \in \operatorname {Aut} (F)$. We reduce the problem to $T$ $p$-primary of finite rank, where in the one case $T$ is injective, and in the other case $T$ is reduced. Suppose $T = \Pi _{i = 1}^m{T_i}$. In our main theorems we prove that in each case these equivalence classes of extensions are in 1-1 correspondence with the equivalence classes of $n$-generated subgroups of $E$ where $E = \Pi _{i = 1}^m{E_i},{E_i} = \operatorname {End} ({T_i})$. Two $n$-generated subgroups of $E$ will be called equivalent if one can be mapped onto the other by an automorphism of $E$.

Note on extensions of abelian groups by primary groups

Note on extensions of abelian groups by primary groups

The dimension subgroup problem

Whether the terms of the lower central series of a group are associated with the powers of the augmentation ideal of its integral group ring is a question which has been open for over thirty years. This question, the dimension subgroup problem, has achieved notoriety for the number of false proofs it has elicited. In this paper, we record general observations on the problem, verify the conjecture for several new classes of groups and develop generalizations which indicate new lines of research. In a subsequent paper [20], we examine the subgroups attached to group rings over arbitrary coefficient domains. The general properties of dimension subgroups are set forth in the first section. Several equivalent formulations of the conjecture are given. The subgroups are calculated in a number of special cases. A minimal counterexample is a finite p-group with cyclic center. In the second section, we take the approach that such reductions make available. The conjecture is established for Abelian-by-cyclic groups and for split extensions of Abelian groups by groups satisfying the conjecture; it follows inductively that the Sylow subgroups of the symmetric groups satisfy the problem and so every p-group embeds in a p-group for which the conjecture is verified. This analysis provides an unexpected bridge to the work of Passi who conjectured a strong result about the properties of factor sets of extensions of divisible groups byp-groups and showed how it would establish the dimension subgroup conjecture. Here another way of using Passi’s conjecture to solve the problem is introduced and applied in a minimal counterexample to obtain new results. The third section places the problem in a wider context: I f G, is associated with I”, then G, is associated with even larger ideals. It may be that these ideals will have to be identified before the dimension subgroup problem can be resolved affirmatively. Some such ideals are discovered by expanding the methods of Passi and Moran. The results obtained are used to establish the

Dimensionality and the Duals of Certain Locally Compact Groups

Let $G$ be a separable locally compact group containing a closed normal subgroup which is type I and regularly embedded. The elements of the dual space $G$ are known to be induced from certain closed subgroups. In this article we first determine the kernel of an induced representation and then give necessary and sufficient conditions for $G$ to be maximally almost periodic. These results are then applied to the collection of compact extensions of abelian groups.

Dimensionality and the duals of certain locally compact groups

Let G G be a separable locally compact group containing a closed normal subgroup which is type I and regularly embedded. The elements of the dual space G G are known to be induced from certain closed subgroups. In this article we first determine the kernel of an induced representation and then give necessary and sufficient conditions for G G to be maximally almost periodic. These results are then applied to the collection of compact extensions of abelian groups.

Compact abelian group extensions of discrete dynamical systems

This paper introduces the notion of a free G extension of a dynamical system where G is a compact abelian group. The concept is closely allied to that of generalised discrete spectrum (which includes Abramov's quasi-discrete spectrum as a special case). We give necessary and sufficient conditions for a G extension of a minimal (uniquely ergodic) dynamical system to be minimal (uniquely ergodic) and show that in a certain sense a general G extension lifts these properties. Stable G-extensions always lift these properties if the underlying space is connected. This fact is then used to characterise all uniquely ergodic and minimal affine transformations of a certain three dimensional nilmanifold. The rest of the paper is devoted to the exhibition of group invariants for systems with generalised discrete spectrum. In particular it is shown that such systems always have a compact abelian group as underlying space. A lemma which facilitates this result gives necessary and sufficient conditions for a connected G-extension of a compact abelian group to be a compact abelian group.

Расширения абелевых групп при помощи групп периодических

Расширения абелевых групп при помощи групп периодических

High extensions of Abelian groups

High extensions of Abelian groups

A note on Abelian group extensions

A note on Abelian group extensions

Finite Extensions of Abelian Groups With Minimum Condition

Finite Extensions of Abelian Groups With Minimum Condition

Finite extensions of Abelian groups with minimum condition

Among the groups with minimum condition one meets, often quite unexpectedly, groups which satisfy one of the following two conditions: (a) the commutator subgroup is finite; (b) there exists an abelian subgroup of finite index. It is our objective in this investigation to give various characterizations of these two classes of groups some of which have very little obvious connection with either the minimum condition or properties (a) and (b). Our principal result, stated in ?0, contains characterizations of the groups with minimum condition and finite commutator subgroup; and the proof of this theorem is effected in ?1 to ?5. In ?6 we specialize this result to show that torsion groups with finite automorphism groups are finite. In ?7 we enunciate a characterization of the groups with minimum condition possessing abelian subgroups of finite index; and the proof of this proposition will be effected in ?8 to ?11. These results are used in ?12 to show that the p-Sylow subgroups of these groups are conjugate. 0. In this section we enunciate and discuss our principal