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Hereditarily minimal topological groups

Hereditarily minimal topological groups AbstractWe study locally compact groups having all subgroups minimal. We call such groups hereditarily minimal.In 1972 Prodanov proved that the infinite hereditarily minimal compact abelian groups are precisely the groups ℤp{\mathbb{Z}_{p}}of p-adic integers.We extend Prodanov’s theorem to the non-abelian case at several levels.For infinite hypercentral (in particular, nilpotent) locally compact groups, we show that the hereditarily minimal ones remain the same as in the abelian case.On the other hand, we classify completely the locally compact solvable hereditarily minimal groups, showing that, in particular, they are always compact and metabelian.The proofs involve the (hereditarily) locally minimal groups, introduced similarly.In particular, we prove a conjecture by He, Xiao and the first two authors, showing that the group ℚp⋊ℚp*{\mathbb{Q}_{p}\rtimes\mathbb{Q}_{p}^{*}}is hereditarily locally minimal, where ℚp*{\mathbb{Q}_{p}^{*}}is the multiplicative group of non-zero p-adic numbers acting on the first component by multiplication.Furthermore, it turns out that the locally compact solvable hereditarily minimal groups are closely related to this group. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Hereditarily minimal topological groups

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Publisher
de Gruyter
Copyright
© 2019 Walter de Gruyter GmbH, Berlin/Boston
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/forum-2018-0066
Publisher site
See Article on Publisher Site

Abstract

AbstractWe study locally compact groups having all subgroups minimal. We call such groups hereditarily minimal.In 1972 Prodanov proved that the infinite hereditarily minimal compact abelian groups are precisely the groups ℤp{\mathbb{Z}_{p}}of p-adic integers.We extend Prodanov’s theorem to the non-abelian case at several levels.For infinite hypercentral (in particular, nilpotent) locally compact groups, we show that the hereditarily minimal ones remain the same as in the abelian case.On the other hand, we classify completely the locally compact solvable hereditarily minimal groups, showing that, in particular, they are always compact and metabelian.The proofs involve the (hereditarily) locally minimal groups, introduced similarly.In particular, we prove a conjecture by He, Xiao and the first two authors, showing that the group ℚp⋊ℚp*{\mathbb{Q}_{p}\rtimes\mathbb{Q}_{p}^{*}}is hereditarily locally minimal, where ℚp*{\mathbb{Q}_{p}^{*}}is the multiplicative group of non-zero p-adic numbers acting on the first component by multiplication.Furthermore, it turns out that the locally compact solvable hereditarily minimal groups are closely related to this group.

Journal

Forum Mathematicumde Gruyter

Published: May 1, 2019

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