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For any locally compact abelian periodic group A its automorphism group contains as a subgroup those automorphisms that leave invariant every closed subgroup of A, to be denoted by SAut(A). This subgroup is again a locally compact abelian periodic group in its natural topology and hence allows a decomposition into its p-primary subgroups for p the primes for which topological p-elements in this automorphism subgroup exist. The interplay between the p-primary decomposition of SAut(A) and A can be encoded in a bipartite graph, the mastergraph of A. Properties and applications of this concept are discussed., Oberwolfach Preprints;2018,05

A locally compact abelian group is called periodic if it is totally disconnected and is a directed union of its compact subgroups. Various aspects of abelian periodic groups are considered such as – decomposing them into local products of their Sylow p- subgroups, – providing new descriptions of periodic abelian torsion groups, and of – periodic abelian divisible groups and their torsion-free and their torsion components, – reviewing splitting theorems, notably for finite rank pure subgroups of (almost) finite exponent p-groups, – providing a definition of a general p-rank for all locally compact abelian p-groups.

We give a detailed description of all totally disconnected locally compact groups with only compact elements in which any two closed subgroups are permutable and their product being a closed subgroup again. This amounts to sharpening the classification results in [2, Section 14] while not being dependent upon them.

Let G be a locally compact topological group, G 0 the connected component of its identity element, and comp ( G ) the union of all compact subgroups. A topological group will be called inductively monothetic if any subgroup generated (as a topological group) by finitely many elements is generated (as a topological group) by a single element. The space SUB ( G ) of all closed subgroups of G carries a compact Hausdorff topology called the Chabauty topology. Let F 1 ( G ) , respectively, R 1 ( G ) , denote the subspace of all discrete subgroups isomorphic to Z , respectively, all subgroups isomorphic to R . It is shown that a necessary and sufficient condition for G ∈ F 1 ( G ) ‾ to hold is that G is Abelian, and either that G ≅ R × comp ( G ) and G / G 0 is inductively monothetic, or else that G is discrete and isomorphic to a subgroup of Q . It is further shown that a necessary and sufficient condition for G ∈ R 1 ( G ) ‾ to hold is that G ≅ R × C for a compact connected Abelian group C .

Locally compact abelian groups are classified in which the sum of any two closed subgroups is itself closed. This amounts to reproving and extending results by Yu.~N.~Mukhin from 1970. Namely we contribute a complete classification of all totally disconnected \lca\ groups with $X+Y$ closed for any closed subgroups $X$ and $Y$., Comment: 38 pages, 1 figure

ZusammenfassungWir skizzieren die Konturen einer Theorie der Halbgruppen, also eines Zweiges der Mathematik, der sich ganz im Gegensatz zu der im 19. Jahrhundert geschaffenen Gruppentheorie im 20. Jahrhundert entwickelte. Als Arbeitshypothese schlagen wir vor, daß die Anfänge der Halbgruppentheorie bereits im Werk von Sophus Lie angelegt sind. Über eine präzise Definition des Gruppenbegriffs war man sich im 19. Jahrhundert noch keineswegs im Klaren. Objekte, die man später als Halbgruppen bezeichnen wird, heißen damals Gruppen. Lie ist sich des Unterschieds wohl bewußt und versucht, sich mit diesem Problem auseinan-derzusetzen. Wir schließen mit einem Ausblick auf die gegenwärtige Forschung über die Lieschen Fundamentalsätze im Rahmen der Halbgruppentheorie.AbstractAfter framing the outlines of a theory of semigroups as a mathematical domain developed in the 20th century in contrast with group theory as an achievement of the 19th, we propose, as a working hypothesis, the claim that the beginning of semigroup theory is indicated in the work of Sophus Lie. Insecurity on the axiomatic definition of a group is still widespread in the nineteenth century; objects which later will be recognized in their own right as semigroups are called groups at that period. Lie is conscious of the difference and attempts to deal with it. We give an outlook on contemporary research on Lie's Fundamental Theorems in the context of semigroups.

Hofmann, K.H. and S.A. Morris, Free compact groups IV: Splitting the component and the structure of the commutator group, Journal of Pure and Applied Algebra 70 (1991) 89-96.

We show that every locally compact group G contains a discrete subspace X which is closed in G ⧹{1} and has the property that G is the smallest closed subgroup containing X . The minimum of the set of all cardinals card X for these X is called the generating rank s ( G ) of G . We show that s(G)≤w(G)≤s(G) ℵ 0 for all G which are not monothetic. We calculate s ( G ) for G with w ( G )≤ c . For compact groups G with w ( G )≤ c and finitely many components, s ( G ) is finite.

’ Technische Hochschule Darmstadt, Schlossgartenstr. 7, D-6100 Darrnstadt, Germany, FRG and Tulane University, New Orleans, La. 70118, U.S.A. 2 Louisiana State University, Baton Rouge, La. 70803, U.S.A. Communicated by Prof. W.T. van Est at the meeting of February 27, 1984

As the second step in understanding the structure of free compact groups, the center Z of a free compact group FX on a compact space X is here discussed. In particular, we shall show that Z is contained in the identity component ( FX ) 0 and that Z is related to the free compact abelian group F a X on X , which group we studied extensively in the first of this series of articles. Indeed, if F ' X denotes the closure of the commutator subgroup of FX , it is shown here that the function z ↦ ( FX ) : Z 0 → F a X = FX / F ' X is a projective cover (in a sense previously specified); its kernel Z 0 ⌢F'X has the character group ⊙ Q /⊙ Z ⊗ H 1 ( X ,⊙ Z ). If X is connected and H 1 ( X ,⊙ Z ) is divisible, then the free compact group is the direct product of its commutator subgroup and the free compact abelian group.