scholarly journals Abelian p-groups with minimal full inertia

2021 ◽  
Author(s):  
Brendan Goldsmith ◽  
Luigi Salce
Keyword(s):  
Endomorphism Ring ◽  
Jacobson Radical ◽  
Structural Condition ◽  
Closure Properties ◽  
Direct Sums ◽  
Cyclic Groups ◽  
Topological Condition ◽  
Fully Invariant Subgroups

AbstractThe class of abelian p-groups with minimal full inertia, that is, satisfying the property that fully inert subgroups are commensurable with fully invariant subgroups is investigated, as well as the class of groups not satisfying this property; it is known that both the class of direct sums of cyclic groups and that of torsion-complete groups are of the first type. It is proved that groups with “small" endomorphism ring do not satisfy the property and concrete examples of them are provided via Corner’s realization theorems. Closure properties with respect to direct sums of the two classes of groups are also studied. A topological condition of the socle and a structural condition of the Jacobson radical of the endomorphism ring of a p-group G, both of which are satisfied by direct sums of cyclic groups and by torsion-complete groups, are shown to be independent of the property of having minimal full inertia. The new examples of fully inert subgroups, which are proved not to be commensurable with fully invariant subgroups, are shown not to be uniformly fully inert.

scholarly journals Subgroups of finite direct sums of valuated cyclic groups

1988 ◽  
Vol 114 (1) ◽  
pp. 1-15 ◽  
Author(s):  
David M Arnold ◽  
Fred Richman
Keyword(s):  
Direct Sums ◽  
Cyclic Groups

Criteria for Total Projectivity

1981 ◽  
Vol 33 (4) ◽  
pp. 817-825 ◽  
Author(s):  
Paul Hill
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Abelian Groups ◽  
General Criterion ◽  
Direct Sums ◽  
Primary Abelian Group ◽  
Cyclic Groups ◽  
Direct Sum ◽  
Totally Projective Groups ◽  
Torsion Groups

All groups herein are assumed to be abelian. It was not until the 1940's that it was known that a subgroup of an infinite direct sum of finite cyclic groups is again a direct sum of cyclics. This result rests on a general criterion due to Kulikov [7] for a primary abelian group to be a direct sum of cyclic groups. If G is p-primary, Kulikov's criterion presupposes that G has no elements (other than zero) having infinite p-height. For such a group G, the criterion is simply that G be the union of an ascending sequence of subgroups Hn where the heights of the elements of Hn computed in G are bounded by some positive integer λ(n). The theory of abelian groups has now developed to the point that totally projective groups currently play much the same role, at least in the theory of torsion groups, that direct sums of cyclic groups and countable groups played in combination prior to the discovery of totally projective groups and their structure beginning with a paper by R. Nunke [11] in 1967.

Sur les Sous-Groupes Purifiables des Groupes Abéliens Primaires

1990 ◽  
Vol 33 (1) ◽  
pp. 11-17 ◽  
Author(s):  
K. Benabdallah ◽  
C. Piché
Keyword(s):  
Abelian Groups ◽  
Divisible Group ◽  
Direct Sums ◽  
Cyclic Subgroups ◽  
Cyclic Groups ◽  
Direct Sum

AbstractThe class of primary abelian groups whose subsocles are purifiable is not yet completely characterized and it contains the class of direct sums of cyclic groups and torsion complete groups. In sharp constrast with this, the class of groups whose p2-bounded subgroups are purifiable consist only of those groups which are the direct sum of a bounded and a divisible group. Various tools are developed and a short application to the pure envelopes of cyclic subgroups is given in the last section.

Quasi-isomorphism of direct sums of cyclic groups

1965 ◽  
Vol 16 (1-2) ◽  
pp. 33-36 ◽  
Author(s):  
R. A. Beaumont ◽  
R. S. Pierce
Keyword(s):  
Direct Sums ◽  
Cyclic Groups

A dual approach to Černikov modules

1977 ◽  
Vol 82 (2) ◽  
pp. 215-239 ◽  
Author(s):  
B. Hartley
Keyword(s):  
Endomorphism Ring ◽  
Finite Rank ◽  
Additive Group ◽  
Main Theme ◽  
Integral Group ◽  
Dual Approach ◽  
Cyclic Groups ◽  
Direct Sum ◽  
Černikov Group ◽  
Torsion Modules

Let G be a group and A a right G-module. If the additive group A+ of A is a Černikov group, that is, a direct sum of finitely many cyclic and quasi-cyclic groups, we shall call A a Černikov module over G or over the integral group ring . Suppose that A+ is, furthermore, a divisible p-group, where p is a prime. Since the endomorphism ring of a quasi-cyclic p-group is isomorphic to the ring of p-adic integers, we find that is a free -module of finite rank. We can make A* into a right G-module in the usual way, and since A* is actually just the Pontrjagin dual of A, Pontrjagin duality shows that A → A* gives rise to a contravariant equivalence between the categories of divisible Černikov p-torsion modules over and G-modules which are -free of finite rank. Since the latter category is to some extent familiar, at least when G is finite – for its objects determine representations of G over the field of p-adic numbers, a field of characteristic zero – we may hope to exploit this correspondence systematically to study divisible Černikov p-modules. This is our main theme.

scholarly journals Pure-complete subgroups of direct sums of Prüfer groups

1972 ◽  
Vol 13 (1) ◽  
pp. 47-48 ◽  
Author(s):  
Paul Hill
Keyword(s):  
Abelian Group ◽  
Pure Subgroup ◽  
Direct Sums ◽  
Primary Abelian Group ◽  
Cyclic Groups ◽  
Direct Sum

Suppose that G is a p-primary abelian group. The subgroup G[p] = {x∈G:px=0} is called the socle of G and any subgroup S of G[p] is called a subsocle of G. If each subsocle of G supports a pure subgroup, then G is said to be pure-complete [1]. It is well known that, if G a direct sum of cyclic groups, then G is necessarily pure-complete. Further results about pure-complete groups are contained in [1] and [3].

Sign in / Sign up

Export Citation Format

Share Document