On the Socles of Fully Inert Subgroups of Abelian p-Groups

Chekhlov, A. R.; Danchev, Peter V.; Goldsmith, Brendan

doi:10.1007/s00009-021-01747-z

Mediterr. J. Math.

2021

DOI: 10.1007/s00009-021-01747-z

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On the Socles of Fully Inert Subgroups of Abelian p-Groups

A. R. Chekhlov

Peter V. Danchev

Brendan Goldsmith

Abstract: A famous conjecture states that any uniformly fully inert subgroup of a given group is commensurable with a fully invariant subgroup (see [3] and [4]). In this short note, we settle this problem completely in the affirmative for an arbitrary Abelian group.

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Cited by 6 publications

References 14 publications

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Abelian p-groups with minimal full inertia

Goldsmith

Salce

2021

Period Math Hung

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The 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.

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Abelian p-groups with minimal full inertia

Goldsmith

Salce

2021

Period Math Hung

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Weakly fully and characteristically inert socle-regular Abelian p-groups

Chekhlov

Danchev

2022

Communications in Algebra

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On the Socles of Strongly Inert Subgroups of Abelian $ p $-Groups

Chekhlov

Danchev²

2023

Sib Math J

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On the Socles of Fully Inert Subgroups of Abelian p-Groups

Abstract: A famous conjecture states that any uniformly fully inert subgroup of a given group is commensurable with a fully invariant subgroup (see [3] and [4]). In this short note, we settle this problem completely in the affirmative for an arbitrary Abelian group.

Cited by 6 publications

References 14 publications

Abelian p-groups with minimal full inertia

Abelian p-groups with minimal full inertia

Weakly fully and characteristically inert socle-regular Abelian p-groups

On the Socles of Strongly Inert Subgroups of Abelian $ p $-Groups

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