Compact groups with probabilistically central monothetic subgroups

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.

We say that an element g of a group G is almost right Engel if there is a finite set R(g) such that for every x∈G, there is a positive integer n(x,g) such that […[[g,x],x],…,x]∈R(g) if x is repeated at least n(x,g) times. Thus, g is a right Engel element precisely when we can choose R(g)={1}. We prove that if all elements of a compact (Hausdorff) group G are almost right Engel, then G has a finite normal subgroup N such that G/N is locally nilpotent. If in addition there is a uniform bound ∣R(g)∣⩽m for the orders of the corresponding sets, then the subgroup N can be chosen of order bounded in terms of m. The proofs use the Wilson–Zelmanov theorem saying that profinite Engel groups are locally nilpotent and previous results of the authors about compact groups in which all elements are almost left Engel.

Suppose that G is a locally compact group, that \({\Gamma}\) is a discrete, finitely generated group, and that $$\varphi\colon G \longrightarrow \Gamma$$ is an ‘abstract’ surjective homomorphism. We are interested in conditions which imply that \({\varphi}\) is automatically continuous. We obtain a complete answer to this question in the case where G is a topologically finitely generated locally compact abelian group or an almost connected Lie group. In these two cases the well-known structure theory for such groups G leads quickly to a solution. The question becomes much more difficult if one assumes only that G is a locally compact group. This leads to interesting questions about normal subgroups in infinite products and in ultraproducts. Łos’ theorem, the solution of the 5th Hilbert problem, and recent results by Nikolov–Segal can be combined to answer the question.

The aim of this paper is to prove certain characterization theorems for groups in which permutability is a transitive relation, the so called $${\mathcal{PT}}$$ -groups. In particular, it is shown that the finite solvable $${\mathcal{PT}}$$ -groups, the finite solvable groups in which every subnormal subgroup of defect two is permutable, the finite solvable groups in which every normal subgroup is permutable sensitive, and the finite solvable groups in which conjugate-permutability and permutability coincide are all one and the same class. This follows from our main result which says that the finite modular p-groups, p a prime, are those p-groups in which every subnormal subgroup of defect two is permutable or, equivalently, in which every normal subgroup is permutable sensitive. However, there exist finite insolvable groups which are not $${\mathcal{PT}}$$ -groups but all subnormal subgroups of defect two are permutable.

Most of the classical metric spaces with metric d have the property that given any two pairs of points a, , a2 , bi, b2 of the space with d(al , a2) = d(b1, b2), there is an isometry of the space carrying a1, a2 to b1, b2 respectively. This property was called two-point homogeneity by Birkhoff [2] and called the (*)-property by Busemann. For simplicity, we shall call such a metric space a (*)-space. Concerning these spaces, very few properties are known even under the further assumption that they are Riemannian. Busemann conjectured that if they are S. L. [6] and closed (i.e., each geodesic is congruent to the euclidean circle), then they are elliptic [6, p. 233]. In this paper, we shall determine all the compact and connected (*)-spaces. It is to be noted that whereas the compact and connected metrically homogeneous spaz es (in the sense of van Dantzig) can be quite pathological such as solenoids and infinite-dimensional spaces, the connected and compact (*)-spaces are found to be closed manifolds with very simple homology properties. In these discussions, we make extensive use of the theory of transformation groups developed by Montgomery and Zippin. In fact, let E be a compact and connected (*)-space. The group G of all its isometries is compact. We shall first show that each non-discrete normal subgroup of G is transitive on E so that G has no arbitrarily small normal subgroup. Thus G is a compact Lie group and E a closed manifold. When the dimension of E is odd, a Riemannian metric can be introduced to render E a Clifford-Klein space form. When the dimension of E is even, the situation becomes more complicated. It will be proved that there exists an isometry of E which has only a finite number of fixed points and the index of each fixed point is equal to unity. It follows then that E has nonvanishing Euler characteristic. Using this property, the underlying topological space of E can be determined. We shall show that a connected, compact (*)space is homeomorphic with either a sphere Sn , a real projective space pn' a. complex projective space K, a quaternion projective space Q4f or the Cayley projective plane M. Conversely, the spheres and the various projective spaces are really the underlying topological spaces of certain (*)-spaces. In the case of Sn and pfnl this is well known. In the case of K2n, Q4' and M'6, we shall construct a convex metric for each of them to render them two-point homogeneous. The resulting metric spaces will be called the complex elliptic spaces, the quaternion elliptic spaces and the Cayley elliptic plane respectively. Let E be a (*)-space with metric d. Suppose d* to be an equivalent metric related with d by an equation of the form: d*(x, y) = f(d(x, y)), x, y e E, where f is a function in a single real variable. The space E with the new metric 177

We study steady-state bifurcation in reversible-equivariant vector fields. We assume an action on the phase space of a compact Lie group G with a normal subgroup H of index two, and study vector fields that are H-equivariant and have all elements of the complement G∖H as time-reversal symmetries.We focus on separable bifurcation problems that can be reduced to equivariant steady-state bifurcation problems, possibly with parameter symmetries. We describe both bifurcations of equilibria that arise when external parameters are varied, and branching of families of equilibria that may arise in the phase space when external parameters are fixed. We also show how our results apply to bifurcation problems for reversible relative equilibria and reversible (relative) periodic orbits.

PROOF. Replacing G by a finite covering we may and will assume that G is a direct product of simple Lie groups, none of which is compact. The group In of isometries of the Euclidean n-space is known to be the semidirect product of the orthogonal group and R , RI being the normal subgroup. Therefore the projection of In onto the orthogonal group is a differentiable group morphism. Let f be a differentiable morphism from G into In. Then f followed by the projection of In onto the orthogonal group is a differentiable morphism. If this composition were nontrivial it would be nontrivial on some simple factor of G. Therefore, there would be a noncompact simple Lie group contained in the orthogonal group. This is impossible because a semisimple Lie subgroup of a compact Lie group is closed, hence compact (cf. [2, p. 128D. Thus, the image of f is contained in RW. This implies that G has a nontrivial abelian factor, unless f is the trivial morphism.

For a locally compact quantum group $\mathbb{G}$ we define its center, $\mathscr{Z}(\mathbb{G})$, and its quantum group of inner automorphisms, $\mathrm{Inn}(\mathbb{G})$. We show that one obtains a natural isomorphism between $\mathrm{Inn}(\mathbb{G})$ and $\mathbb{G}/\!\mathscr{Z}(\mathbb{G})$, we characterize normal quantum subgroups of a compact quantum group as those left invariant by the action of the quantum group of inner automorphisms and discuss several examples.

The Burnside ring A{G) of an arbitrary compact group G is defined in analogy to torn Dieck's definition for a compact Lie group.A topology is introduced on the set of conjugacy classes of closed subgroups, and a theorem of Dress on the topology of the prime spectrum of A(G) for profinite G is improved and extended to this general setting.This paper is condensed from part of my doctoral thesis [5], written under the supervision of Professor T. torn Dieck, to whom I am indebted for encouragement and advice.1.The Burnside ring and its additive structure.Suppose G is a compact group, and define a G-manifold as a differentiable manifold M together with a continuous left G-action such that, for each g G G, the translation g : M - M is a diffeomorphism.Examples are real G-representations and manifolds invariantly imbedded in such.Theorem 1.1.If G is a compact group, M a compact effective G-manifold, then G is a Lie group and M is a differentiable G-manifold.Proof.Since M is compact and locally connected, it has finitely many connected components, say M = Ji"A/,.For 1 < / < n, set G, = {g G GI g(M) = A/,}.This is a subgroup of G, and, since for any w, G M, the map /: G- M via f(g) = g-m is continuous, while G, = f~x(M), G, is clopen in G. Thus H = n"G, is a clopen normal subgroup.Now set H = {h E H\im G M, h-m = m], so that H is a normal subgroup of H and H/H acts effectively on A/,.Then since M is connected, Montgomery-Zippin [7, p. 208, Theorem 2] shows /////, to be a Lie group.Hence H/C\"xHi is again a Lie group.But since G operates effectively.Pi?//, = {h G //|Vw G M, h m = m) = {e}, so that H is a Lie group, and with it G since it is open in G.That M is then a differentiable G-manifold follows from [7, p. 208, Theorem 3].Now for any G-space X, let t(X) = {g G G|Vx G X,g x = x) be that part of G which acts trivially.Then for a G-manifold M, the G-action is derived from an effective G/i(A/)-action via the projection G-G/i(A/).Hence G/t(M) is a Lie group.

Hertwecks counterexample demonstrates that the normalizer property need not necessarily hold for a nite group even if the normalizer property hods both for a normal subgroup and the corresponding quotient group of it. In this paper, we shall prove the following main result. Let G be a nite group with a normal subgroup N such that all central units of Z( G / N ) are trivial. If the Sylow 2-subgroup of N is a direct product factor of N , then the normalize property holds for G .

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.

1·1. We shall write for the class of groups all of whose non-trivial normal subgroups have finite index. Thus, rather obviously, finite groups, simple groups, and the infinite cyclic and dihedral groups all lie in the class . Other examples of -groups are to be found in a variety of contexts. The main result of Mennicke (8) shows that, for n ≥ 3, the factor group of SL(n, Z) by its centre is a -group, where SL(n, Z) denotes the unimodular group of all n × n invertible matrices with integer coefficients and with determinant 1. In McLain(7), an example is given of an infinite, locally finite, locally soluble -group, whose only non-trivial normal subgroups are the terms of its derived series, and in (4), P. Hall discusses an infinite -group, all of whose proper quotients are finite p-groups. If is a class of groups closed under taking homomorphic images, it is easily seen that the existence of an infinite -group satisfying the maximal condition for normal subgroups implies the existence of an infinite -group in the class , so that certain questions concerning the finiteness of groups satisfying the maximal condition for normal subgroups can be interpreted as questions about -groups.

Let G G be a locally compact group P ( G ) = { 0 ≦ ϕ ∈ L 1 ( G ) ; ∫ ϕ ( x ) d x = 1 } P(G) = \{ 0 \leqq \phi \in {L_1}(G);\int {\phi (x)dx = 1\} } and ( l a f ) ( x ) = a f ( x ) = f ( a x ) ({l_a}f)(x) = {}_af(x) = f(ax) for all a , x ∈ G a,x \in G and f ∈ L ∞ ( G ) .0 ≦ Ψ ∈ L ∞ ( G ) ∗ , Ψ ( 1 ) = 1 f \in {L^\infty }(G).0 \leqq \Psi \in {L^\infty }{(G)^ \ast },\Psi (1) = 1 is said to be a [topological] left invariant mean ([TLIM] LIM) if Ψ ( a f ) = Ψ ( f ) [ Ψ ( ϕ ∗ f ) = Ψ ( f ) \Psi {{\text {(}}_a}f) = \Psi (f)[\Psi (\phi \ast f) = \Psi (f) ] for all a ∈ G , ϕ ∈ P ( G ) , f ∈ L ∞ ( G ) a \in G,\phi \in P(G),f \in {L^\infty }(G) . The main result of this paper is the Theorem. Let G G be a locally compact group, amenable as a discrete group. If G G contains an open σ \sigma -compact normal subgroup, then LIM = TLIM if and only if G G is discrete. In particular if G G is an infinite compact amenable as discrete group then there exists some Ψ ∈ L I M \Psi \in LIM which is different from normalized Haar measure. A harmonic analysis type interpretation of this and related results are given at the end of this paper . 2 ^{2}

We are concerned with conditions under which a locally compact group $G$ has a maximal compact normal subgroup $K$ and whether or not $G/K$ is a Lie group. If $G$ has small compact normal subgroups $K$ such that $G/K$ is a Lie group, then $G$ is pro-Lie. If in $G$ there is a collection of closed normal subgroups $\{ {H_\alpha }\}$ such that $\cap {H_\alpha } = e$ and $G/{H_\alpha }$ is a Lie group for each $\alpha$, then $G$ is a residual Lie group. We determine conditions under which a residual Lie group is pro-Lie and give an example of a residual Lie group which is not embeddable in a pro-Lie group.

Bernstein's theorem for compact groups