2-affine complete algebras need not be affine complete

For each we exhibit a finite algebra R _k such that R _k is k-affine complete, but not (k+1)-affine complete; this means that every k-ary congruence preserving function on R _k lies in , but there is a (k +1)-ary congruence preserving function of R _k that does not lie in . Received September 27, 2001; accepted in final form February 9, 2002.     

Über die affin vollständigen,endlich erzeugbaren Moduln

A universal algebraA is calledk-affine complete, if any function of the Cartesian powerA ^k intoA, which is compatible with all congruence relations ofA, is a polynomial function.A is called affine complete, if it isk-affine complete for every integerk. In this paper, all affine complete finitely generated modules are characterized. Moreover, the paper contains some results on functions compatible with all congruence relations of an algebra, and on affine complete algebras in general.

---

---

Herrn Prof. Dr. E. Hlawka zum 60. Geburtstag gewidmet     

On Hagemann's and Herrmann's characterization of strictly affine complete algebras

In [4], J.Hagemann and C. Herrmann have characterized those algebras that are strictly k-affine complete for all . In the present note, we prove a very similar characterization using the term condition commutator. Received June 15, 1999; accepted in final form March 3, 2000.     

两个正规p-超可解子群的积

本文研究了两个正规的p-超可解子群的积构成的极小非P-超可解群的结构的问题.利用有限群论和群类论的一些基本方法,获得了两个正规的p-超可解子群的积仍为P-超可解群的一些充分条件,并推广了文[1]中关于超可解群的情况.     

1-affine completeness of compatible modules

A faithful compatible module G of a nearring R is 1-affine complete if R equals the set of congruence preserving functions of G that are 0-preserving. The study of 1-affine complete expanded groups G can be approached from this perspective by considering G as a module for the nearring of 0-preserving polynomial functions of G. In this paper, we obtain some strong conditions on the minimal R-ideals of G, when R has dccr and G is 1-affine complete, by using results from papers on units of such nearrings. We then go on to apply these results to obtain necessary and sufficient conditions for G to have a composition series in which each factor of G by a series term is 1-affine complete.     

On Discrete Zariski-Dense Subgroups of Algebraic Groups

We investigate under which conditions an algebraic group G defined over a locally compact field k admitr a subgroup Γ? G(k) which is dense in the Zariski topology, but discerte in the topology induced by the locally compact topology on k. For non—solvable groups we provide a complete answer.     

A remark on divisibility of definable groups

We show that if G is a definably compact, definably connected definable group defined in an arbitrary o‐minimal structure, then G is divisible. Furthermore, if G is defined in an o‐minimal expansion of a field, k ∈ ? and p_k : G → G is the definable map given by p_k (x ) = x^k for all x ∈ G , then we have |(p_k )^–1(x )| ≥ k^r for all x ∈ G , where r > 0 is the maximal dimension of abelian definable subgroups of G . (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

k-Involutions of SL(n,k) over fields of characteristic 2

Symmetric k-varieties generalize Riemannian symmetric spaces to reductive groups defined over arbitrary fields. For most perfect fields, it is known that symmetric k-varieties are in one-to-one correspondence with isomorphy classes of k-involutions. Therefore, it is useful to have representatives of each isomorphy class in order to describe the k-varieties. Here we give matrix representatives for each isomorphy class of k-involutions of SL(n,k) in the case that k is any field of characteristic 2; we also describe fixed-point groups of each type of involution.     

Infinite Classes of Dihedral Snarks

Flower snarks and Goldberg snarks are two infinite families of cyclically 5–edge–connected cubic graphs with girth at least five and chromatic index four. For any odd integer k, k > 3, there is a Flower snark, say J _k , of order 4k and a Goldberg snark, say B _k , of order 8k. We determine the automorphism groups of J _k and B _k for every k and prove that they are isomorphic to the dihedral group D _4k of order 4k. Research performed within the activity of INdAM–GNSAGA with the financial support of the Italian Ministry MIUR, project “Strutture Geometriche, Combinatoria e loro Applicazioni”.     

Ehresmann connection for the canonical foliation on a manifold over a local algebra

For a canonical foliation on a manifoldM ^A over a local algebra, theA-affine horizontal distribution complementary to the leaves, similar to the horizontal distribution of a higher order connection on the fiber bundle ofA-jets, is defined. In the case of a complete manifoldM ^A, theA-affine horizontal distribution is proved to be an Ehresmann connection in the sense of Blumental-Hebda. It is shown that theA-affine horizontal distribution onM ^A exists if and only if the Atiyah class of a certain foliated principal bundle vanishes.Translated fromMatematicheskie Zametki, Vol. 59, No. 2, pp. 303–310, February, 1996.     

<Emphasis Type="Italic">k</Emphasis>-Free-like groups

The following results are proved. In Theorem 1, it is stated that there exist both finitely presented and not finitely presented 2-generated nonfree groups which are k-free-like for any k ⩾ 2. In Theorem 2, it is claimed that every nonvirtually cyclic (resp., noncyclic and torsion-free) hyperbolic m-generated group is k-free-like for every k ⩾ m + 1 (resp., k ⩾ m). Finally, Theorem 3 asserts that there exists a 2-generated periodic group G which is k-free-like for every k ⩾ 3. Supported by NSF (grant Nos. DMS 0455881 and DMS-0700811). (A. Yu. Olshanskii, M. V. Sapir) Supported by RFBR project No. 08-01-00573. (A. Yu. Olshanskii) Supported by BSF grant (USA–Israel). (M. V. Sapir) Translated from Algebra i Logika, Vol. 48, No. 2, pp. 245–257, March–April, 2009.     

A decomposition theorem for G-groups

Some classical results about linear representations of a finite group G have been also proved for representations of G on non-abelian groups （G-groups）. In this paper we establish a decomposition theorem for irreducible G-groups which expresses a suitable irreducible G-group as a tensor product of two projective G-groups in a similar way to the celebrated theorem of Clifford for linear representations. Moreover, we study the non-abelian minimal normal subgroups of G in which this decomposition is possible.     

Parametrization of tamely ramified maximal tori using bounded subgroups

Let be a reductive group defined over a local complete field F with discrete valuation, and split over some unramified extension of F, and let G be its group of F-points. In this paper, we define a class of abelian “torus-like” subgroups in nonreductive groups, called pseudo-tori, which generalizes the notion of torus, and we establish a correspondence between conjugacy classes of tamely ramified maximal tori of G and association classes of maximal pseudo-tori of the quotients of parahorics of G by their second congruence subgroup, viewed as groups of k-points of algebraic groups defined over the residual field k of F.      

Nonsoluble length of finite groups with restrictions on Sylow subgroups

By 𝔛(n) we denote the variety of all groups satisfying the law [x,y]ⁿ≡1, that is, groups with commutators of order dividing n. Let p be a prime and G a finite group whose Sylow p-subgroups have normal series of length k all of whose quotients belong to 𝔛(n). We show that the non-p-soluble length λ_p(G) of G is bounded in terms of k and n only (Theorem 1.2). In the case where p is odd, a stronger result is obtained (Theorem 1.3).     

The L p -curvature images of convex bodies and L p -projection bodies

Associated with the L _p -curvature image defined by Lutwak, some inequalities for extended mixed p-affine surface areas of convex bodies and the support functions of L _p -projection bodies are established. As a natural extension of a result due to Lutwak, an L _p -type affine isoperimetric inequality, whose special cases are L _p -Busemann-Petty centroid inequality and L _p -affine projection inequality, respectively, is established. Some L _p -mixed volume inequalities involving L _p -projection bodies are also established.     

Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields II

Let k be a separably closed field. Let G be a reductive algebraic k-group. We study Serre’s notion of complete reducibility of subgroups of G over k. In particular, using the recently proved center conjecture of Tits, we show that the centralizer of a k-subgroup H of G is G-completely reducible over k if it is reductive and H is G-completely reducible over k. We show that a regular reductive k-subgroup of G is G-completely reducible over k. We present examples where the number of overgroups of irreducible subgroups and the number of G(k)-conjugacy classes of k-anisotropic unipotent elements are infinite.     

On Groups Admitting a Fixed-Point-Free Elementary 2-Group of Automorphisms

We give a short proof that if G is a finite group of derived length k and if G admits a fixed-point-free action of the elementary group of order 2 ⁿ , then G has a normal series of length n all of whose quotients are nilpotent of class bounded in terms of k and n only.     

Reconstruction of Function Fields

For k an algebraic closure of the finite field , ℓ prime distinct from p and X a surface over k, we prove that the field of rational functions k(X) can be recovered from the maximal pro-ℓ-quotient of its absolute Galois group – in fact already from the second central descending series quotient of . Submitted: July 2004, Revision: October 2005, Final revision: February 2008, Accepted: February 2008     

Generalized symmetric elements generated by a prime sequence

Let be a prime sequence in a local Noether lattice L. For denotes the set of finite joins in L of power products of the generalized symmetric elements of order k (majorization elements) in together with 0 and I. We have previously showed that for is a Noetherian distributive -domain. For and for any is again such a sub--domain of . For and is not closed under the meet of . However with its induced meet is again a Noetherian distributive -domain. Each finite set of majorization elements asymptotically forms a distributive sublattice of for k sufficiently large. Received March 2, 1998; accepted in final form June 11, 1998.     

Right distributive quasigroups on algebraic varieties

In this paper we develop a structure theory of algebraic right distributive quasigroups which correspond to closed and connected conjugacy classes generating algebraic Fischer groups (in the sense of [6]) such that the mappingx x ^–1 ax, fora , is an automorphism of (as variety). We also give examples of algebraic Fischer groups where this does not happen. It becomes clear that the class of algebraic right distributive quasigroups has nice properties concerning subquasigroups, normal subquasigroups and direct product.We give a complete classification of one- and two-dimensional as well as of minimal algebraic right distributive quasigroups.