On categories with semidirect products

Necessary and sufficient conditions for a pointed category to admit semidirect products, in the sense of Bourn and Janelidze (1998) [3], are provided and interpreted in terms of protomodularity and exactness of appropriate split chains.     

On discrete frames associated with semidirect products

For groups which are the semidirect product of some vector group with a unimodular group we prove that the existence of a discrete frame obtained from an at-most countable set of vectors through the action of a given unitary representation implies that the representation in use has to be square-integrable.     

On the hyperdecidability of semidirect products of pseudovarieties

The notion of hyperdecidability has been introduced as a tool which is particularly suited for granting decidability of semidirect products. It is shown in this paper that the semidirect product of an hyperdecidable pseudovariety with a pseudovariety whose finitely generated free objects are finite and effectively computable is again hyperdecidable. As instances of this result, one obtains, for example, the hyperdecidability of the pseudovarieties of ail finite completely simple semigroups and of all finite bands of left groups.     

On semidirect products and wreath products of partially ordered groups

The notions of Cartesian and semidirect products for partially ordered groups are considered. A series of results on those products of AO mathcal{A}mathcal{O} -groups and interpolation groups is obtained. Some results concerning wreath products of directed groups are obtained.     

Subgroups of semidirect products

The notion of a reduced crossed homomorphism is introduced and subgroups of a semidirect product are described by means of it. Subsemidirect products and semidirect products with a given structure of normal subgroups are characterized.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 1048–1055, July–August, 1991.     

Weyl quantization for semidirect products

Let G be the semidirect product V?K where K is a connected semisimple non-compact Lie group acting linearily on a finite-dimensional real vector space V. Let O be a coadjoint orbit of G associated by the Kirillov-Kostant method of orbits with a unitary irreducible representation π of G. We consider the case when the corresponding little group K₀ is a maximal compact subgroup of K. We realize the representation π on a Hilbert space of functions on Rn where n=dim(K)−dim(K₀). By dequantizing π we then construct a symplectomorphism between the orbit O and the product R2n×O^′ where O^′ is a little group coadjoint orbit. This allows us to obtain a Weyl correspondence on O which is adapted to the representation π in the sense of [B. Cahen, Quantification d'une orbite massive d'un groupe de Poincaré généralisé, C. R. Acad. Sci. Paris Série I 325 (1997) 803-806]. In particular we recover well-known results for the Poincaré group.     

Complete finite semidirect products and wreath products

A complete group is one with a trivial center and with all automorphisms inner. This paper uses group cohomology to give a sufficient condition for a finite semidirect product G = N \rtimes H{G = N \rtimes H} with C _G (N) ≤ N to be complete and proves a partial converse. These results are enough to fully characterize complete finite permutational wreath products and to specialize that characterization in the case of finite standard wreath products.     

Hecke algebras of semidirect products

We consider group-subgroup pairs in which the group is a semidirect product and the subgroup is contained in the normal part. We give conditions for the pair to be a Hecke pair and we show that the enveloping Hecke algebra and Hecke -algebra are canonically isomorphic to semigroup crossed products, generalizing earlier results of Arledge, Laca and Raeburn and of Brenken.

     

Automorphism groups of semidirect products

This paper shows that if is a semidirect product of finite groups, then if and only if and for all . As an application, we investigate the automorphism group of a split metacyclic p-group for odd p. The second author is supported by the Natural Science Foundation of China (10671058).     

Orthodox semidirect products and wreath products of monoids

Communicated by H.-J. Hoehnke     

On the asymptotic behaviour of convolution powers of probabilities on discrete groups

We give examples to show that a discrete group can carry two symmetric finitely supported probabilities , each satisfying a local limit law µ^*n (e~C ⁿ n ^–, but for different 's. This disproves a conjecture ofGerl [3].     

A note on orthodox semidirect products and wreath products of monoids

In the present note, we show that the restriction which Saito imposed on S in [3] is not necessary: S may be a semigroup without unity. Secondly, we investigate the wreath products of semigroups and obtain our main results, Theroem 2.3 and Corollary 2.4. We find also that groups in [3] may be replaced by right groups in [1]. Undefined terminology and notation may be found in [3] and [1] unlesss otherwise stated.     

半环半直积的同构定理

半群半直积及封闭性刻划于文[1],为将半直积的研究推广至半环中,本文首先在交换半环与交换半群的直积上定义加法及乘法运算,从而构造了一类半环;半环半直积．并讨论了半环半直积的同构等问题．     

Bilateral semidirect products of transformation semigroups

Communicated by B. M. Schein