Periodic locally compact groups. A study of a class of totally disconnected topological groups.

*(English)*Zbl 1423.22001
De Gruyter Studies in Mathematics 71. Berlin: De Gruyter (ISBN 978-3-11-059847-6/hbk; 978-3-11-059919-0/ebook). liii, 301 p. (2019).

The book collects recent research of the authors on periodic locally compact groups and represents a substantial enrichment of the literature on locally compact (abelian) groups. It also re-evaluates some aspects of locally compact abelian groups that are in the literature but not easily accessible. The treatise is completed with the necessary background and references, and with a high number of useful examples. It is aimed to experts in the field with a strong background in the theory of locally compact groups and in group theory.

The fundamental class of locally compact groups for this book is that of near abelian groups. A topological group \(G\) is inductively monothetic if for every finite subset \(F\) of \(G\) there exists \(g\in G\) such that \(\overline{\langle F\rangle}=\overline{\langle g\rangle}\). A topological group \(G\) is near abelian if \(G\) is locally compact and contains a closed normal abelian subgroup \(A\), called base of \(G\), such that every closed subgroup of \(A\) is normal in \(G\) and \(G/A\) is abelian and inductively monothetic. Clearly, a near abelian group is metabelian.

The structure of near abelian groups is studied from several points of view and with different techniques. In particular, it turns out that if \(G\) is a near abelian group and \(A\) a base of \(G\), then \(G\) is totally disconnected and \(A\) is totally disconnected and compactly covered (i.e., every element of \(A\) is contained in a compact subgroup of \(A\)). From here comes the relevance of the concept which gives the title to the book: a topological group \(G\) is periodic if \(G\) is a totally disconnected and compactly covered locally compact group.

The book consists of three parts: background on locally compact groups, near abelian groups, applications. They are preceded by a very useful overview of the contents of the book, starting with a historical review of the genesis of locally compact groups from Hilbert’s fifth problem. Moreover, the authors explain here their motivations as well as the scope of the book, they recall the main results and their mutual relations, together with their connections to results already existing in the literature.

After a necessary background on totally disconnected locally compact groups, the Chabauty space associated to every locally compact group is studied; this is the family of all closed subgroups of the group endowed with a suitable compact Hausdorff topology.

Periodic groups are shown to possess many noteworthy links with classical group theory and a generalization to periodic groups is presented of the Sylow theory for profinite groups; in particular, one can find a generalization of the Schur-Zassenhaus theorem, a fixed-point theorem and a generalized version of the Maschke theorem.

A large portion of the book, especially in the first part, is dedicated to the abelian case and in particular to the structure of locally compact abelian groups. Even if the structure of such groups is widely known, several new interesting results are proposed here for periodic abelian groups. Special attention is paid to the case of locally compact abelian \(p\)-groups, for which a new concept of \(p\)-rank is given. The notion of scalar automorphism is introduced for periodic abelian groups and the Sylow structure of the group of scalar automorphisms is investigated using also its associated prime graph.

A classification theorem for inductively monothetic locally compact groups is shown, moreover a classification of periodic inductively monothetic groups is used in a structure theorem for periodic near abelian groups. Near abelian groups are further studied in all details by applying results and techniques developed in the first part of the book. Furthermore, the prime graph of a periodic near abelian group is introduced and it turns out to be a powerful tool in the investigation of the Sylow structure of the periodic near abelian group. A whole chapter is dedicated to examples of near abelian groups.

Recall that a topological group \(G\) is topologically quasihamiltonian if \(\overline{XY}=\overline{YX}\) for every pair \(X,Y\) of closed subgroups of \(G\), while \(G\) is strongly topologically quasihamiltonian if for every pair \(X,Y\) of closed subgroups of \(G\), also \(XY\) is a closed subgroup of \(G\). Clearly, every strongly topologically quasihamiltonian group is topologically quasihamiltonian but the converse does not hold. Moreover, \(G\) is topologically modular if the lattice of all closed subgroups of \(G\) is modular; every strongly topologically quasihamiltonian group is topologically modular, but also in this case the latter class is strictly larger.

As an application of the results in the previous part of the book, a classification of topologically quasihamiltonian periodic groups is provided, starting from the case of \(p\)-groups; in particular every topologically quasihamiltonian locally compact \(p\)-group is near abelian. Moreover, the structure of topologically modular periodic groups is investigated and several results are presented, also in relation to periodic and near abelian groups. Finally, a classification of strongly topologically quasihamiltonian periodic abelian groups is presented, and given \(G\) a non-periodic locally compact group, \(G\) is shown to be strongly topologically quasihamiltonian precisely when \(G\) is near abelian with a base that is strongly topologically quasihamiltonian and torsion, equivalently \(G\) is topologically quasihamiltonian and topologically modular.

The interested reader can go back to the papers of the authors [Topology Appl. 263, 26–43 (2019; Zbl 1442.22006); “A study in locally compact groups – Chabauty space, Sylow theory, the Schur-Zassenhaus formalism, the prime graph for near abelian groups”, Commun. Stoch. Anal. 10, No. 4, 515–540 (2016; doi:10.31390/cosa.10.4.09); “When is the sum of two closed subgroups closed in a locally compact abelian group?”, Topology Appl. (to appear)]. There one can find the research results presented here, and in particular the latter one contains several improvements. See also the Errata by the authors at the link https://www.asc.tuwien.ac.at/~herfort/HHR_BOOK/errata.pdf.

The fundamental class of locally compact groups for this book is that of near abelian groups. A topological group \(G\) is inductively monothetic if for every finite subset \(F\) of \(G\) there exists \(g\in G\) such that \(\overline{\langle F\rangle}=\overline{\langle g\rangle}\). A topological group \(G\) is near abelian if \(G\) is locally compact and contains a closed normal abelian subgroup \(A\), called base of \(G\), such that every closed subgroup of \(A\) is normal in \(G\) and \(G/A\) is abelian and inductively monothetic. Clearly, a near abelian group is metabelian.

The structure of near abelian groups is studied from several points of view and with different techniques. In particular, it turns out that if \(G\) is a near abelian group and \(A\) a base of \(G\), then \(G\) is totally disconnected and \(A\) is totally disconnected and compactly covered (i.e., every element of \(A\) is contained in a compact subgroup of \(A\)). From here comes the relevance of the concept which gives the title to the book: a topological group \(G\) is periodic if \(G\) is a totally disconnected and compactly covered locally compact group.

The book consists of three parts: background on locally compact groups, near abelian groups, applications. They are preceded by a very useful overview of the contents of the book, starting with a historical review of the genesis of locally compact groups from Hilbert’s fifth problem. Moreover, the authors explain here their motivations as well as the scope of the book, they recall the main results and their mutual relations, together with their connections to results already existing in the literature.

After a necessary background on totally disconnected locally compact groups, the Chabauty space associated to every locally compact group is studied; this is the family of all closed subgroups of the group endowed with a suitable compact Hausdorff topology.

Periodic groups are shown to possess many noteworthy links with classical group theory and a generalization to periodic groups is presented of the Sylow theory for profinite groups; in particular, one can find a generalization of the Schur-Zassenhaus theorem, a fixed-point theorem and a generalized version of the Maschke theorem.

A large portion of the book, especially in the first part, is dedicated to the abelian case and in particular to the structure of locally compact abelian groups. Even if the structure of such groups is widely known, several new interesting results are proposed here for periodic abelian groups. Special attention is paid to the case of locally compact abelian \(p\)-groups, for which a new concept of \(p\)-rank is given. The notion of scalar automorphism is introduced for periodic abelian groups and the Sylow structure of the group of scalar automorphisms is investigated using also its associated prime graph.

A classification theorem for inductively monothetic locally compact groups is shown, moreover a classification of periodic inductively monothetic groups is used in a structure theorem for periodic near abelian groups. Near abelian groups are further studied in all details by applying results and techniques developed in the first part of the book. Furthermore, the prime graph of a periodic near abelian group is introduced and it turns out to be a powerful tool in the investigation of the Sylow structure of the periodic near abelian group. A whole chapter is dedicated to examples of near abelian groups.

Recall that a topological group \(G\) is topologically quasihamiltonian if \(\overline{XY}=\overline{YX}\) for every pair \(X,Y\) of closed subgroups of \(G\), while \(G\) is strongly topologically quasihamiltonian if for every pair \(X,Y\) of closed subgroups of \(G\), also \(XY\) is a closed subgroup of \(G\). Clearly, every strongly topologically quasihamiltonian group is topologically quasihamiltonian but the converse does not hold. Moreover, \(G\) is topologically modular if the lattice of all closed subgroups of \(G\) is modular; every strongly topologically quasihamiltonian group is topologically modular, but also in this case the latter class is strictly larger.

As an application of the results in the previous part of the book, a classification of topologically quasihamiltonian periodic groups is provided, starting from the case of \(p\)-groups; in particular every topologically quasihamiltonian locally compact \(p\)-group is near abelian. Moreover, the structure of topologically modular periodic groups is investigated and several results are presented, also in relation to periodic and near abelian groups. Finally, a classification of strongly topologically quasihamiltonian periodic abelian groups is presented, and given \(G\) a non-periodic locally compact group, \(G\) is shown to be strongly topologically quasihamiltonian precisely when \(G\) is near abelian with a base that is strongly topologically quasihamiltonian and torsion, equivalently \(G\) is topologically quasihamiltonian and topologically modular.

The interested reader can go back to the papers of the authors [Topology Appl. 263, 26–43 (2019; Zbl 1442.22006); “A study in locally compact groups – Chabauty space, Sylow theory, the Schur-Zassenhaus formalism, the prime graph for near abelian groups”, Commun. Stoch. Anal. 10, No. 4, 515–540 (2016; doi:10.31390/cosa.10.4.09); “When is the sum of two closed subgroups closed in a locally compact abelian group?”, Topology Appl. (to appear)]. There one can find the research results presented here, and in particular the latter one contains several improvements. See also the Errata by the authors at the link https://www.asc.tuwien.ac.at/~herfort/HHR_BOOK/errata.pdf.

Reviewer: Anna Giordano Bruno (Udine)

##### MSC:

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |

22B05 | General properties and structure of LCA groups |