Maximal Protori in Compact Topological Groups

This report aims at giving a general overview on the classification of the maximal subgroups of compact Lie groups (not necessarily connected). In the first part, it is shown that these fall naturally into three types: (1) those of trivial type, which are simply defined as inverse images of maximal subgroups of the corresponding component group under the canonical projection and whose classification constitutes a problem in finite group theory, (2) those of normal type, whose connected one-component is a normal subgroup, and (3) those of normalizer type, which are the normalizers of their own connected one-component. It is also shown how to reduce the classification of maximal subgroups of the last two types to: (2) the classification of the finite maximal Σ-invariant subgroups of centerfree connected compact simple Lie groups and (3) the classification of the Σ-primitive subalgebras of compact simple Lie algebras, where Σ is a subgroup of the corresponding outer automorphism group. In the second part, we explicitly compute the normalizers of the primitive subalgebras of the compact classical Lie algebras (in the corresponding classical groups), thus arriving at the complete classification of all (non-discrete) maximal subgroups of the compact classical Lie groups. Mathematics Subject Classification 2000: 22E15.

Mathematical Proceedings of the Cambridge Philosophical Society, 2001

Colloquium Mathematicum

Transactions of The American Mathematical Society, 1984

In [1] Smidt's conjecture on the existence of an infinite abelian subgroup in any infinite group is settled by counterexample. The well-known Hall-Kulatilaka Theorem asserts the existence of an infinite abelian subgroup in any infinite locally finite group. This paper discusses a topological analogue of the problem. The simultaneous consideration of a stronger condition-that centralizers of nontrivial elements be compact-turns out to be useful and, in essence, inevitable. Thus two compactness conditions that give rise to a profinite arithmetization of topological groups are added to the classical list (see, e.g., [13 or 4]).

Ukrainian Mathematical Journal, 1990

Proceedings of the American Mathematical Society, 1973

In this paper it is shown that if A" is a torus-like continuum, then X has the shape of a compact connected abelian topological group. Let IT be a collection of compact connected Lie groups. In light of the above result it is natural to ask if a LT-like continuum has the shape of a compact connected topological group. An example is given to show that this is not the case.

2010

We show that every compact connected group is the limit of a continuous inverse sequence, in the category of compact groups, where each successor bonding map is either an epimorphism with finite kernel or the projection from a product by a simple compact Lie group. As an application, we present a proof of an unpublished result of Charles Mills from 1978: every compact group is supercompact.

Axioms

Here “group” means additive abelian group. A compact group G contains δ–subgroups, that is, compact totally disconnected subgroups Δ such that G/Δ is a torus. The canonical subgroup Δ(G) of G that is the sum of all δ–subgroups of G turns out to have striking properties. Lewis, Loth and Mader obtained a comprehensive description of Δ(G) when considering only finite dimensional connected groups, but even for these, new and improved results are obtained here. For a compact group G, we prove the following: Δ(G) contains tor(G), is a dense, zero-dimensional subgroup of G containing every closed totally disconnected subgroup of G, and G/Δ(G) is torsion-free and divisible; Δ(G) is a functorial subgroup of G, it determines G up to topological isomorphism, and it leads to a “canonical” resolution theorem for G. The subgroup Δ(G) appeared before in the literature as td(G) motivated by completely different considerations. We survey and extend earlier results. It is shown that td, as a functor,...

Central European Journal of Mathematics, 2010

Abstract: We show that every compact connected group is the limit of a continuous inverse sequence, in the category of compact groups, where each successor bonding map is either an epimorphism with finite kernel or the projection from a product by a simple compact Lie ...

Journal of the Australian Mathematical Society, 1967

In this paper a number of questions about locally compact groups are studied. The structure of finite dimensional connected locally compact groups is investigated, and a fairly simple representation of such groups is obtained. Using this it is proved that finite dimensional arcwise connected locally compact groups are Lie groups, and that in general arcwise connected locally compact groups are locally connected. Semi-simple locally compact groups are then investigated, and it is shown that under suitable restrictions these satisfy many of the properties of semi-simple Lie groups. For example, a factor group of a semi-simple locally compact group is semi-simple. A result of Zassenhaus, Auslander and Wang is reformulated, and in this new formulation it is shown to be true under more general conditions. This fact is used in the study of (C)-groups in the sense of K. Iwasawa.