A structure theorem on compact groups

In 2004, Fang, Lu, Wang and Xu proved that, with few exceptions, connected Cayley digraphs of finite nonabelian simple groups with out-valency two are normal, enabling a complete determination of the automorphism groups of such Cayley digraphs. In this paper, we present significant improvements: we not only reduce the list of exceptions but also extend these results to the more general setting of products of nonabelian simple groups. This includes the case of nonabelian characteristically simple groups, which is crucial in Praeger’s approach to analyzing the automorphism group of a general Cayley digraph.

On square roots of the Haar state on compact quantum groups

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.

Let $\\alpha $ be an automorphism of a finite group $G$. For a positive integer $n$, let $E_{G,n}(\\alpha )$ be the subgroup generated by all commutators $[...[[x,\\alpha ], \\alpha ],\\dots ,\\alpha ]$ in the semidirect product $G\\langle\\alpha \\rangle$ over $x\\in G$, where $\\alpha $ is repeated $n$ times. By Baer's theorem, if $E_{G,n}(\\alpha )=1$, then the commutator subgroup $[G,\\alpha ]$ is nilpotent. We generalize this theorem in terms of certain length parameters of $E_{G,n}(\\alpha )$. For soluble $G$ we prove that if, for some $n$, the Fitting height of $E_{G,n}(\\alpha )$ is equal to $k$, then the Fitting height of $[G,\\alpha ]$ is at most $k+1$. For nonsoluble $G$ the results are in terms of the nonsoluble length and generalized Fitting height. The generalized Fitting height $h^*(H)$ of a finite group $H$ is the least number $h$ such that $F^*_h(H)=H$, where $F^*_0(H)=1$, and $F^*_{i+1}(H)$ is the inverse image of the generalized Fitting subgroup $F^*(H/F^*_{i}(H))$. Let $m$ be the number of prime factors of the order $|\\alpha |$ counting multiplicities. It is proved that if, for some $n$, the generalized Fitting height of $E_{G,n}(\\alpha )$ is equal to $k$, then the generalized Fitting height of $[G,\\alpha ]$ is bounded in terms of $k$ and $m$. 
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\nThe nonsoluble length~$\\lambda (H)$ of a finite group~$H$ is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if $\\lambda (E_{G,n}(\\alpha ))=k$, then the nonsoluble length of $[G,\\alpha ]$ is bounded in terms of $k$ and $m$.
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\nWe also state conjectures of stronger results independent of $m$ and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups.

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.

If K is a closed subgroup of a compact group G, the probability that a randomly chosen pair of elements from K and G commute is denoted by Pr(K, G). Say that a subgroup K ≤ G is ϵ-central in G if Pr(〈g〉, G) ≥ ϵ for any g in K. Here 〈g〉 denotes the monothetic subgroup generated by g ∈ G. Our main result is that if K is ϵ-central in G, then there is an ϵ-bounded number e and a normal subgroup T ≤ G such that both the index [G:T] and the order of the commutator subgroup [Ke, T] are finite and ϵ-bounded. In particular, if G is a compact group for which there is ϵ > 0 such that Pr(〈g〉, G) ≥ ϵ for any g ∈ G, then there is an ϵ-bounded number e and a normal subgroup T such that both the index [G:T] and the order of [Ge, T] are finite and ϵ-bounded.

We say that a compact Lie group G has the Borsuk-Ulam property in the weak sense if for every orthogonal representation V of G and every G-equivariant map /: S{V) -> S(V), VG = {0}, of the unit sphere we have deg/ Φ 0 . We say that G has the Borsuk-Ulam property in the strong sense if for any two orthogonal representations V, W of G with dim W = dim V and WG = VG = {0} and every G-equivariant map /: S(V) —> S{W) of the unit spheres we have deg/ Φ 0. In this paper a complete classification, up to isomorphism, of group with the weak Borsuk-Ulam property is given. A classification of groups with the strong Borsuk-Ulam property does not cover nonabelian p-groups with all elements of the order p . In fact we deal with a more general definition admitting a nonempty fixed point set of G on the sphere S(V). 1. The main theorems. In order to formulate our main results we introduce the following notation. Let G be a compact Lie group. We denote by Go the component of identity of and by Γ the quotient group G/GQ . We use standard notation of the theory of compact transformation groups (see for instance [4] or [5]). In particular, for every subgroup H c G, the fixed point set of H on a G-space X is denoted by XH. Also, for a G-equivariant map f:X—>Y between two G-spaces, we denote by fH its restriction to the space XH . The symbol (n, m) stands for the greatest common divisor of the integers n, m with the notation (0, 1) = 0 and \G for the rank of the (finite) group G. We will work with the following definition of the BorsukUlam property (cf. [11]). DEFINITION I. (A) We say that G has the Borsuk-Ulam property in the weak sense A if for every orthogonal representation V of G and every G-equivariant map /: S(V) -> S(V) if (deg/ G, |Γ|) = 1 then

We survey recent results regarding embeddings of finite simple groups (and their nonsplit central extensions) in complex Lie groups, especially the Lie groups of exceptional type. Introduction Throughout this paper, L will be a finite group. Representation theory for L is usually understood to be the study of group morphisms L → GL(n, k) for distinguished collections of fields k (e.g., all overfields of a fixed field F ) and positive integers n . The topic of this survey is motivated by the question as to what happens if GL ( n ,·) is replaced by another algebraic group G (·). We shall mainly be concerned with the case where L is a finite simple group (that is, a finite nonabelian simple group) or a central extension thereof, and G(k) is a connected simple algebraic group over a field k . A further restriction of our discussion concerns the field k . It will mostly be taken to be the complex numbers, in which case we will mainly study group morphisms from L to the complex Lie group G (ℂ). (See below for some exceptions in §3 and §5.) For G (·) of classical type, the theory for representations L → G (ℂ) differs little from the usual one for GL ( n , ℂ). Indeed, a representation L → GL ( n , ℂ) decomposes into irreducible subrepresentations. The decomposition is well controlled by character theory.

The set π(G) of all prime divisors of the order of a finite group G is often called its prime spectrum. It is proved that every finite simple nonabelian group G has sections H1, …, Hm of some special form such that π(H1)∪…∪π(Hm) = π(G) and m ≤ 5. Moreover, m ≤ 2 if G is an alternating or classical simple group. In all cases, it is possible to choose the sections Hi so that each of them is a simple nonabelian group, a Frobenius group, or (in one case) a dihedral group. If the above equality holds for a finite group G, then we say that the set {H1,…,Hm} controls the prime spectrum of G. We also study some parameter c(G) of finite groups G related to the notion of control.

We review the notion of simple compact quantum groups and examples, and discuss the problem of construction and classification of simple compact quantum groups. Several new quantum groups constructed by Banica, Curran and Speicher since the author's first paper on simple quantum groups are shown to be simple using results of Raum and Weber.

We classify elementary abelian 2-subgroups of compact simple Lie groups of adjoint type. This finishes the classification of elementary abelian \(p\)-subgroups of compact simple Lie groups (equivalently, complex linear algebraic simple groups) of adjoint type started in Griess (Geom Dedicata 39(3):253–305, 1991).

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.

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.

Hardy’s uncertainty principle for the Gabor transform is proved for locally compact abelian groups having noncompact identity component and groups of the form$\mathbb{R}^{n}\times K$, where$K$is a compact group having irreducible representations of bounded dimension. We also show that Hardy’s theorem fails for a connected nilpotent Lie group$G$which admits a square integrable irreducible representation. Further, a similar conclusion is made for groups of the form$G\times D$, where$D$is a discrete group.

A ‐tuple of core‐free subgroups of a finite group is said to be regular if has a regular orbit on the Cartesian product . The regularity number of , denoted by , is the smallest positive integer with the property that every such ‐tuple is regular. In this paper, we develop some general methods for studying the regularity of subgroup tuples in arbitrary finite groups, and we determine the precise regularity number of all almost simple groups with an alternating or sporadic socle. For example, we prove that and . We also formulate and investigate natural generalisations of several well‐studied problems on base sizes for finite permutation groups, including conjectures due to Cameron, Pyber and Vdovin. For instance, we extend earlier work of Burness, O'Brien and Wilson by proving that for every almost simple sporadic group, with equality if and only if is the Mathieu group . We also show that every triple of soluble subgroups in an almost simple sporadic group is regular, which generalises recent work of Burness on base sizes for transitive actions of sporadic groups with soluble point stabilisers.