In a recent article [K. H. Hofmann and F. G. Russo, ‘The probability that $x$ and $y$ commute in a compact group’, Math. Proc. Cambridge Phil Soc., to appear] we calculated for a compact group $G$ the probability $d(G)$ that two randomly selected elements $x, y\in G$ satisfy $xy=yx$, and we discussed the remarkable consequences on the structure of $G$ which follow from the assumption that $d(G)$ is positive. In this note we consider two natural numbers $m$ and $n$ and the probability $d_{m,n}(G)$ that for two randomly selected elements $x, y\in G$ the relation $x^my^n=y^nx^m$ holds. The situation is more complicated whenever $n,m\gt 1$. If $G$ is a compact Lie group and if its identity component $G_0$ is abelian, then it follows readily that $d_{m,n}(G)$ is positive. We show here that the following condition suffices for the converse to hold in an arbitrary compact group $G$: for any nonopen closed subgroup $H$ of $G$, the sets $\{g\in G: g^k\in H\}$ for both $k=m$ and $k=n$ have Haar measure $0$. Indeed, we show that if a compact group $G$ satisfies this condition and if $d_{m,n}(G)\gt 0$, then the identity component of $G$is abelian.

We show that a compact group G has finite conjugacy classes, i.e., is an FC-group if and only if its center Z(G) is open if and only if its commutator subgroup G′ is finite. Let d(G) denote the Haar measure of the set of all pairs (x,y) in G×G for which [x,y]=1; this, formally, is the probability that two randomly picked elements commute. We prove that d(G) is always rational and that it is positive if and only if G is an extension of an FC-group by a finite group. This entails that G is abelian by finite. The proofs involve measure theory, transformation groups, Lie theory of arbitrary compact groups, and representation theory of compact groups. Examples and references to the history of the discussion are given at the end of the paper.

A pro-Lie group is a projective limit of finite dimensional Lie groups. It is proved that a surjective continuous group homomorphism between connected pro-Lie groups is open. In fact this remains true for almost connected pro-Lie groups where a topological group is called almost connected if the factor group modulo the identity component is compact. As consequences we get a Closed Graph Theorem and the validity of the Second Isomorphism Theorem for pro-Lie groups in the almost connected context.

Two commuting real matrices $A$ and $B$ have commuting exponentials $\exp A$ and $\exp B$, a fact observed for instance in linear algebra or differential equations courses. The converse implication is false. A clarification of this phenomenon is proposed that makes use of the theory of the exponential function $\exp\colon{\fam\euffam g}\to G$ of a real Lie group $G$ and its singularities. In Section 1, a catalog of low-dimensional examples illustrates various ways that, for two elements $X, Y\in{\fam\euffam g}$, the commuting of $\exp X$ and $\exp Y$ in $G$ may fail to entail the commuting of $X$ and $Y$ in ${\fam\euffam g}$. In Section 2, consequences of the relation $[\exp X,\exp Y]={\bf 1}$ are inspected, whereby certain regularity assumptions on $X$ and $Y$ are made. A regular element $Y$ of the Lie algebra ${\fam\euffam g}$ determines a Cartan subalgebra ${\fam\euffam h}={\fam\euffam g}^0(Y)$ of ${\fam\euffam g}$ and a certain subgroup ${\cal W}_Y$ of the (finite!) Weyl group of ${\fam\euffam g}$ with respect to the Cartan subalgebra ${\fam\euffam h}$. If, additionally, the exponential function is regular at $X$ and at $Y$, then the ordered pair $(X,Y)$ is said to be in general position. If $(X,Y)$ is in general position, then the relation $[\exp X,\exp Y]={\bf 1}$ in $G$ permits the definition of a certain element $w(X,Y)\in{\cal W}_Y$. Let ${\fam\euffam z}({\fam\euffam g})$ denote the center of ${\fam\euffam g}$. It is shown that, if $\exp X$ and $\exp Y$ commute in $G$ for $(X,Y)$ in general position, then $[X,Y]\in{\fam\euffam z}({\fam\euffam g})\cap[{\fam\euffam h},{\fam\euffam h}]$ iff $w(X,Y)={\bf 1}$. Write $H\defi\exp{\fam\euffam h}$, and let $Z(G)$ denote the center of $G$. If the identity component of $Z(G)\cap[H,H]$ is simply connected, and if $\exp X$ and $\exp Y$ commute for $(X,Y)$ in general position, then $[X,Y]=0$ iff $w(X,Y)={\bf 1}$. If $G$ is simply connected compact, then $[\exp X,\exp Y]={\bf 1}$ and $[X,Y]=0$ are equivalent for all pairs $(X,Y)$ in general position. In ${\mathop{\rm SO}\nolimits}(3)$ this is not the case; here $|{\cal W}_Y|=2$. In Section 3, examples show that the validity of the equation $\exp X\exp Y=\exp\!(X+Y)$ has no implications whatsoever in the direction of the commuting of $\exp X$ and $\exp Y$. Finally, in Section 4, it is shown that, for a simply connected Lie group $G$, the commuting of $X, Y\in{\fam\euffam g}$ and that of $\exp X,\exp Y\in G$ are equivalent properties for all$X$ and $Y$ if and only if the exponential function is injective. This class of Lie groups was characterized in terms of other properties by Dixmier and by Saito, independently, in 1957.

For a topological group $G$ we define $\cal N$ to be the set of all normal subgroups modulo which $G$ is a finite-dimensional Lie group. Call $G$ a pro-Lie group if, firstly, $G$ is complete, secondly, $\cal N$ is a filter basis, and thirdly, every identity neighborhood of $G$ contains some member of $\cal N$. It is easy to see that every pro-Lie group $G$ is a projective limit of the projective system of all quotients of $G$ modulo subgroups from $\cal N$. The converse implication emerges as a difficult proposition, but it is shown here that any projective limit of finite-dimensional Lie groups is a pro-Lie group. It is also shown that a closed subgroup of a pro-Lie group is a pro-Lie group, and that for any closed normal subgroup $N$ of a pro-Lie group $G$, for any one parameter subgroup $Y \colon \mathbb{R} \to G/N$ there is a one parameter subgroup $X \colon \mathbb{R}\to G$ such that $X(t) N = Y(t)$ for any real number $t$. The category of all pro-Lie groups and continuous group homomorphisms between them is closed under the formation of all limits in the category of topological groups and the Lie algebra functor on the category of pro-Lie groups preserves all limits and quotients.

In this paper we formulate a new conjecture and introduce methods to verify it in many cases.

We prove a new structure theorem which we call the Countable Layer Theorem. It says that for any compact group G we can construct a countable descending sequence G = Ω0(G) ⊇ … ⊇ Ωn(G) … of closed characteristic subgroups of G with two important properties, namely, that their intersection ∩∞n=1 Ωn(G) is Z0(G0), the identity component of the center of the identity component G0 of G, and that each quotient group Ωn−1(G)/Ωn(G), is a cartesian product of compact simple groups (that is, compact groups having no normal subgroups other than the singleton and the whole group).

In the special case that G is totally disconnected (that is, profinite) the intersection of the sequence is trivial. Thus, even in the case that G is profinite, our theorem sharpens a theorem of Varopoulos [8], who showed in 1964 that each profinite group contains a descending sequence of closed subgroups, each normal in the preceding one, such that each quotient group is a product of finite simple groups. Our construction is functorial in a sense we will make clear in Section 1.

An open subgroup U of a topological group G is always closed, since U is the complement of the open set . An arbitrary closed subgroup C of G is almost never open, unless G belongs to a small family of exceptional groups. In fact, if G is a locally compact abelian group in which every non-trivial subgroup is open, then G is the additive group δp of p-adic integers or the additive group Ωp of p-adic rationale (cf. Robertson and Schreiber[5[, proposition 7). The fact that δp has interesting properties as a topological group has many roots. One is that its character group is the Prüfer group ℤp∞, which makes it unique inside the category of compact abelian groups. But even within the bigger class of not necessarily abelian compact groups the p-adic group δp is distinguished: it is the only one all of whose non-trivial subgroups are isomorphic (cf. Morris and Oates-Williams[2[), and it is also the only one all of whose non-trivial closed subgroups have finite index (cf. Morris, Oates-Williams and Thompson [3[).

It is well-known that a compact connected abelian group G has weight w(G) less than or equal to the cardinality c of the continuum if and only if it is monothetic; that is, if and only if it can be topologically generated by one element. Hofmann and Morris [2] extended this by showing that a compact connected (not necessarily abelian) group can be topologically generated by two elements if and only if w(G) ≤ c.

Trace forms have been well studied as invariant quadratic forms on finite dimensional Lie algebras; the best known of these forms in the Cartan-Killing form. All those forms, however, have the ideal [L, L] ∩ R (with the radical R) in the orthogonal L⊥ and thus are frequently degenerate. In this note we discuss a general construction of Lie algebras equipped with non-degenerate quadratic forms which cannot be obtained by trace forms, and we propose a general structure theorem for Lie algebras supporting a non-degenerate invariant quadratic form. These results complement and extend recent developments of the theory of invariant quadratic forms on Lie algebras by Hilgert and Hofmann [2], keith [4], and Medina and Revoy [7].

A Hausdorff space X is said to be compactly generated (a k-space) if and only if the open subsets U of X are precisely those subsets for which K ∩ U is open in K for all compact subsets of K of X. We interpret this property as a duality property of the lattice O(X) of open sets of X. This view point allows the introduction of the concept of being quasicompactly generated for an arbitrary sober space X. The methods involve the duality theory of up-complete semilattices, and certain inverse limit constructions. In the process, we verify that the new concept agrees with the classical one on Hausdorff spaces.