ON PARTIALLY SATURATED FORMATIONS AND RESIDUALS OF FINITE GROUPS

Partially saturated formations of finite groups are studied with the help of group functions. The results are applied for studying F-groups of automorphisms and π-complements to F-residuals.     

Coleman automorphisms of finite groups and their minimal normal subgroups

In this paper, we show that all Coleman automorphisms of a finite group with self-central minimal non-trivial characteristic subgroup are inner; therefore the normalizer property holds for these groups. Using our methods we show that the holomorph and wreath product of finite simple groups, among others, have no non-inner Coleman automorphisms. As a further application of our theorems, we provide partial answers to questions raised by M. Hertweck and W. Kimmerle. Furthermore, we characterize the Coleman automorphisms of extensions of a finite nilpotent group by a cyclic p-group. Finally, we note that class-preserving Coleman automorphisms of p-power order of some nilpotent-by-nilpotent groups are inner, extending a result by J. Hai and J. Ge, where p is a prime number.     

Irreducible decompositions of unitary representations of infinite- dimensional groups

This paper concerns the problem of irreducible decompositions of unitary representations of topological groups G, including the group Diff₀(M) of diffeomorphisms with compact support on smooth manifolds M. It is well known that the problem is affirmative, when G is a locally compact, separable group (cf. [3, 4]). We extend this result to infinite-dimensional groups with appropriate quasi-invariant measures, and, in particular, we show that every continuous unitary representation of Diff₀(M) has an irreducible decomposition under a fairly mild condition. This research was partially supported by a Grant-in-Aid for Scientific Research (No.14540167), Japan Socieity of the Promotion of Science.     

On Anosov automorphisms of nilmanifolds

The only known examples of Anosov diffeomorphisms are hyperbolic automorphisms of infranilmanifolds, and the existence of such automorphisms is a really strong condition on the rational nilpotent Lie algebra determined by the lattice, so called an Anosov Lie algebra. We prove that n⊕?⊕n (s times, s≥2) has an Anosov rational form for any graded real nilpotent Lie algebra n having a rational form. We also obtain some obstructions for the types of nilpotent Lie algebras allowed, and use the fact that the eigenvalues of the automorphism are algebraic integers (even units) to show that the types (5,3) and (3,3,2) are not possible for Anosov Lie algebras.     

Scale function vs topological entropy

In the realm of topological automorphisms of totally disconnected locally compact groups, the scale function introduced by Willis in [19] is compared with the topological entropy. We prove that the logarithm of the scale function is always dominated by the topological entropy and we provide examples showing that this inequality can be strict. Moreover, we give a condition equivalent to the equality between these two invariants. Various properties of the scale function, inspired by those of the topological entropy, are presented.     

Analogues of Cayley graphs for topological groups

We define for a compactly generated totally disconnected locally compact group a graph, called a rough Cayley graph, that is a quasi-isometry invariant of the group. This graph carries information about the group structure in an analogous way to the ordinary Cayley graph for a finitely generated group. With this construction the machinery of geometric group theory can be applied to topological groups. This is illustrated by a study of groups where the rough Cayley graph has more than one end and a study of groups where the rough Cayley graph has polynomial growth. Supported by project J2245 of the Austrian Science Fund (FWF) and be an IEF Marie Curie Fellowship of the Commission of the European Union.     

Dense orbits of automorphisms and compactness of groups

It is proved, by using topological properties, that when a group automorphism of a locally compact totally disconnected group is ergodic under the Haar measure, the group is compact. The result is an answer for Halmos's question that has remained open for the totally disconnected case.     

THE NUMBER AND SIZE OF ORBITS OF SOME PERMUTATION GROUPS

Abstract

It is shown that Aut ?, the group of homeomorphisms of the rational numbers with the usual topology, has 2 ^N_o orbits on the power set P(?). We call S ? ? a moiety if S and its complement in ? are infinite. It is shown that the orbit of any moiety S under Aut ? has cardinality 2^N_o while the orbit of S under Aut(?, ≤), the group of order preserving automorphisms of ?, has cardinality N_o if and only if S is a finite union of disjoint rational intervals with rational endpoints.     

Trees, wreath products and finite Gelfand pairs

We present a new construction of finite Gelfand pairs by looking at the action of the full automorphism group of a finite spherically homogeneous rooted tree of type r on the variety V(r,s) of all spherically homogeneous subtrees of type s.This generalizes well-known examples as the finite ultrametric space, the Hamming scheme and the Johnson scheme.We also present further generalizations of these classical examples. The first two are based on Harary's notions of composition and exponentiation of group actions. Finally, the generalized Johnson scheme provides the inductive step for the harmonic analysis of our main construction.     

Automorphisms of groups

The survey presents classical assertions due to Nielsen, Whitehead, and others, well-known theorems on automorphisms included in monographs on group theory, and recent results in this area. Attention is focused on the progress in automorphism groups theory for free, solvable, modular, and profinite groups. New tools of investigation using graphs and geometrical ideas are also discussed.     

Dualité dans l’espace des caractères virtuels tempérés d’un groupe réductif <Emphasis Type="Italic">p</Emphasis>-adique

Résumé We define an involution on the set of tempered virtual characters of a connected reductive p-adic group. This involution commutes with character of parabolic induction and with truncation. It also preserves the irreducible characters up to sign and the elliptic inner product.     

Free actions of virtually FC-groups

If an infinite group G admits a free action by a group of automorphisms A which is virtually an FC-group and which has only finitely many orbits, then G is isomorphic to the additive group of a field and the action is that of a group of semilinear transformations. Received: 21 February 2005     

Free subgroups of branch groups

A procedure is described for constructing branch groups on the binary tree, which yields in particular finitely generated branch groups with non-cyclic free subgroups.     

On Minimal Generating Sets of E(D n ), A(D n) and I(D n ) with Even n

Let Dn be the dihedral group of order 2n. Denote by E(Dn) (resp. A(Dn), I(Dn)) the distributively generated nearring generated by the set of all endomorphisms (resp. automorphisms, inner automorphisms). In this paper, we determine for each one of the above three nearrings a minimal (additive) generating set. For E(Dn), this set contains the identity mapping and four other endomorphisms; for A(Dn), the identity mapping, one outer automorphism and one inner automorphisms; and for I(Dn), the identity mapping and two inner automorphisms.     

On pure subgroups of locally compact abelian groups

In this note, we construct an example of a locally compact abelian group G = C × D (where C is a compact group and D is a discrete group) and a closed pure subgroup of G having nonpure annihilator in the Pontrjagin dual $\hat{G}$, answering a question raised by Hartman and Hulanicki. A simple proof of the following result is given: Suppose ${\frak K}$ is a class of locally compact abelian groups such that $G \in {\frak K}$ implies that $\hat{G} \in {\frak K}$ and nG is closed in G for each positive integer n. If H is a closed subgroup of a group $G \in {\frak K}$, then H is topologically pure in G exactly if the annihilator of H is topologically pure in $\hat{G}$. This result extends a theorem of Hartman and Hulanicki.Received: 4 April 2002     

Class-Preserving Coleman Automorphisms of Finite Groups

 Let G be a finite group whose Sylow 2-subgroups are either cyclic, dihedral, or generalized quaternion. It is shown that a class-preserving automorphism of G of order a power of 2 whose restriction to any Sylow subgroup of G equals the restriction of some inner automorphism of G is necessarily an inner automorphism. Interest in such automorphisms arose from the study of the isomorphism problem for integral group rings, see [6, 7, 13, 14]. Received 30 September 2001; in revised form 10 December 2001     

Quasiregular representations of the infinite-dimensional Borel group

The notion of quasiregular (Representation of Lie groups, Nauka, Moscow, 1983) or geometric (Grundlehren der Mathematischen Wissenschaften, Band 220, Springer, Berlin, New York, 1976; Encyclopaedia of Mathematical Science, Vol. 22, Springer, Berlin, 1994, pp. 1-156) representation is well known for locally compact groups. In the present work an analog of the quasiregular representation for the solvable infinite-dimensional Borel group is constructed and a criterion of irreducibility of the constructed representations is presented. This construction uses G-quasi-invariant Gaussian measures on some G-spaces X and extends the method used in Kosyak (Funktsional. Anal. i Prilo?hen 37 (2003) 78-81) for the construction of the quasiregular representations as applied to the nilpotent infinite-dimensional group .