Inertial properties in groups
Let $G$ be a group and $p$ be an endomorphism of $G$. A subgroup $H$ of $G$ is called $p$-inert if $H^pcap H$ has finite index in the image $H^p$. The subgroups that are $p$-inert for all inner automorphisms of $G$ are widely known and studied in the literature, under the name inert subgroup...
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| Format: | Article |
| Language: | English |
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University of Isfahan
2018-09-01
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| Series: | International Journal of Group Theory |
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| Online Access: | http://ijgt.ui.ac.ir/article_21611_00d5ab9d6cd65813b0631a40fa7db9fb.pdf |
| _version_ | 1818158604999458816 |
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| author | Ulderico Dardano Dikran Dikranjan Silvana Rinauro |
| author_facet | Ulderico Dardano Dikran Dikranjan Silvana Rinauro |
| author_sort | Ulderico Dardano |
| collection | DOAJ |
| description | Let $G$ be a group and $p$ be an endomorphism of $G$. A subgroup $H$ of $G$ is called $p$-inert if $H^pcap H$ has finite index in the image $H^p$. The subgroups that are $p$-inert for all inner automorphisms of $G$ are widely known and studied in the literature, under the name inert subgroups. The related notion of inertial endomorphism, namely an endomorphism $p$ such that all subgroups of $G$ are $p$-inert, was introduced in cite{DR1} and thoroughly studied in cite{DR2,DR4}. The ``dual" notion of fully inert subgroup, namely a subgroup that is $p$-inert for all endomorphisms of an abelian group $A$, was introduced in cite{DGSV} and further studied in cite{Ch+, DSZ,GSZ}. The goal of this paper is to give an overview of up-to-date known results, as well as some new ones, and show how some applications of the concept of inert subgroup fit in the same picture even if they arise in different areas of algebra. We survey on classical and recent results on groups whose inner automorphisms are inertial. Moreover, we show how inert subgroups naturally appear in the realm of locally compact topological groups or locally linearly compact topological vector spaces, and can be helpful for the computation of the algebraic entropy of continuous endomorphisms. |
| first_indexed | 2024-12-11T15:32:44Z |
| format | Article |
| id | doaj.art-adc897b673ff4d3484b528213f89173b |
| institution | Directory Open Access Journal |
| issn | 2251-7650 2251-7669 |
| language | English |
| last_indexed | 2024-12-11T15:32:44Z |
| publishDate | 2018-09-01 |
| publisher | University of Isfahan |
| record_format | Article |
| series | International Journal of Group Theory |
| spelling | doaj.art-adc897b673ff4d3484b528213f89173b2022-12-22T01:00:02ZengUniversity of IsfahanInternational Journal of Group Theory2251-76502251-76692018-09-0173176210.22108/ijgt.2017.2161121611Inertial properties in groupsUlderico Dardano0Dikran Dikranjan1Silvana Rinauro2Dipartimento Matematica e Appl., v. Cintia, M.S.Angelo 5a, I-80126 Napoli (Italy)Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, 33100 Udine, Italy.Silvana Rinauro, Dipartimento di Matematica, Informatica ed Economia, Universit`a della Basilicata, Via dell’Ateneo Lucano 10, I-85100 Potenza, Italy.Let $G$ be a group and $p$ be an endomorphism of $G$. A subgroup $H$ of $G$ is called $p$-inert if $H^pcap H$ has finite index in the image $H^p$. The subgroups that are $p$-inert for all inner automorphisms of $G$ are widely known and studied in the literature, under the name inert subgroups. The related notion of inertial endomorphism, namely an endomorphism $p$ such that all subgroups of $G$ are $p$-inert, was introduced in cite{DR1} and thoroughly studied in cite{DR2,DR4}. The ``dual" notion of fully inert subgroup, namely a subgroup that is $p$-inert for all endomorphisms of an abelian group $A$, was introduced in cite{DGSV} and further studied in cite{Ch+, DSZ,GSZ}. The goal of this paper is to give an overview of up-to-date known results, as well as some new ones, and show how some applications of the concept of inert subgroup fit in the same picture even if they arise in different areas of algebra. We survey on classical and recent results on groups whose inner automorphisms are inertial. Moreover, we show how inert subgroups naturally appear in the realm of locally compact topological groups or locally linearly compact topological vector spaces, and can be helpful for the computation of the algebraic entropy of continuous endomorphisms.http://ijgt.ui.ac.ir/article_21611_00d5ab9d6cd65813b0631a40fa7db9fb.pdfcommensurableinertinertial endomorphismentropyintrinsic entropyscale functiongrowthlocally compact grouplocally linearly compact spaceMahler measureLehmer problem |
| spellingShingle | Ulderico Dardano Dikran Dikranjan Silvana Rinauro Inertial properties in groups International Journal of Group Theory commensurable inert inertial endomorphism entropy intrinsic entropy scale function growth locally compact group locally linearly compact space Mahler measure Lehmer problem |
| title | Inertial properties in groups |
| title_full | Inertial properties in groups |
| title_fullStr | Inertial properties in groups |
| title_full_unstemmed | Inertial properties in groups |
| title_short | Inertial properties in groups |
| title_sort | inertial properties in groups |
| topic | commensurable inert inertial endomorphism entropy intrinsic entropy scale function growth locally compact group locally linearly compact space Mahler measure Lehmer problem |
| url | http://ijgt.ui.ac.ir/article_21611_00d5ab9d6cd65813b0631a40fa7db9fb.pdf |
| work_keys_str_mv | AT uldericodardano inertialpropertiesingroups AT dikrandikranjan inertialpropertiesingroups AT silvanarinauro inertialpropertiesingroups |