Arbeitstagung: Gruppen und Topologische Gruppen Vienna July 6 July 7, Abstracts

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1 Arbeitstagung: Gruppen und Topologische Gruppen Vienna July 6 July 7, 202 Abstracts

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3 ENTROPY VS SCALE FUNCTION IN LOCALLY COMPACT GROUPS DIKRAN DIKRANJAN Adler, Konheim and McAndrew [] introduced the topological entropy h top for continuous selfmaps of compact spaces and briefly outlined a possibility to define algebraic entropy for endomorphisms of abstract abelian groups. Later this definition was extended by Peters [3] to automorphisms of arbitrary abelian groups, and recently an extension to arbitrary continuous endomorphisms of locally compact abelian groups was obtained by Virili [4]. Moreover, in the category D of totally disconnected locally compact groups both the algebraic h alg and the topological entropy h top can be defined in a quite simple and uniform way using the fact that they have a local base formed by compact open subgroups. In the abelian case these two entropies are connected by a nice relation provided by Pontryagin duality: h alg (φ) = h top ( φ), where φ denotes the Pontryagin dual of the endomorphism φ. In the decade 994/2004 George Willis [2 4] introduced and studied the scale function for automorphisms in the category D. The definition of these functions uses substantially the fact that for a compact open subgroup U of a group G D the subgroups U and φ(u) are commensurable (i.e., U φ(u) has finite index in both U and φ(u)). Using this idea, intrinsic algebraic entropy for endomorphisms of discrete abelian groups was defined and studied in [2]. The talk will introduce all these notions and will discuss the relations between the various (entropy or scale) functions defined on the morphisms in D. References [] R. L. Adler, A. G. Konheim, M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 4 (965) [2] D. Dikranjan, A. Giordano Bruno, L. Salce, S. Virili, Intrinsic algebraic entropy, submitted. [3] J. Peters, Entropy on discrete Abelian groups, Adv. Math. 33 (979) 3. [4] S. Virili, Entropy for endomorphisms of LCA groups, Topology Appl. 59 (202), [5] G. A. Willis, The structure of totally disconnected, locally compact groups, Math. Ann., 300(2) (994), [6] G. A. Willis, Further properties of the scale function on a totally disconnected group, J. Algebra, 237 (200), [7] G. A. Willis, Tidy subgroups for commutating automorphisms of totally disconnected groups: an analogue of simultaneous triangularisation of matrices, New York J. Math., 0 (2004), -35. Date: Department of Mathematics and Computer Science, University of Udine University of Udine, 206 via delle Scienze, Udine, Italy dikran.dikranjan@uniud.it.

4 FIXED POINTS OF AUTOMORPHISMS IN SIMPLE LOCALLY FINITE GROUPS KIVANÇ ERSOY This is a joint-work with Pavel Shumyatsky. We study the following question: Problem. Does there exist an infinite simple locally finite group G with an elementary abelian p-group of automorphisms A of order p r with r 3 satisfying the following conditions: () C G (A) is finite, (2) For every α A\{}, the subgroup (C G (α)) has finite exponent? In this talk, we will answer this question negatively. In particular, we will prove the following results: Theorem. (E.-Shumyatsky) Let G be a simple locally finite group and A be an elementary abelian p-group of automorphisms of G of order p r such that () C G (A) is finite, (2) There exists some d 0 such that 2 d r and for every α A\{}, the subgroup (C G (α)) (d) has finite exponent. Then G is finite. Theorem 2. (E.-Shumyatsky) Let G be an infinite simple locally finite group and A be an elementary abelian p subgroup of automorphisms of G such that () C G (A) is finite. (2) For every α A\{}, (C G (x)) (d) has finite exponent for some d 2. Then G = P SL p (k) for some infinite locally finite field k of characteristic q p, the subgroup A consists of inner-diagonal automorphisms and A = p 2. Department of Mathematics, Mimar Sinan Fine Arts University Istanbul, Turkey address: kivanc.ersoy@msgsu.edu.tr

5 Multiplication groups of low-dimensional topological loops Ágota Figula Abstract In this talk we discuss the question which Lie groups can occur as the multiplication groups M ult(l) of low-dimensional connected topological loops L and we describe the correspondences between the structure of the group Mult(L) and the structure of the loop L. We prove that a Lie group locally isomorphic to P SL 2 (C) is not the multiplication group of a connected topological loop L. If there exists a connected topological loop L having a Lie group locally isomorphic to SL 3 (R) as its multiplication group, then one has dim L = 5.

6 NAIVE CONFIGURATIONS CHRISTOPH HERING We describe a new way to construct finite geometric objects. For every k we obtain a symmetric configuration E(k ) with k points on a line. In particular, we have a constructive existence proof for such configurations. The method is purely geometric. It also produces interesting periodic matrices. Mathematisches Institut, Universität Tübigen, Deutschland address: christoph.hering@uni-tuebingen.de

7 Sharply transitive sets of permutations Peter Müller Wien, 6./7. Juli 202 Let G be a permutation group acting on a finite set Ω. We develop several criteria on G which show non existence of sharply transitive or sharply 2 transitive subsets of G. In particular, a simple counting argument replaces O Nan s character theoretic technique, thereby extending his results. This is joint work with Gábor Nagy (Szeged).

8 Groups whose prime graph on conjugacy class sizes has few complete vertices E. Pacifici We will discuss some results, recently obtained in a joint work with C. Casolo, S. Dolfi and L. Sanus, concerning the prime graph built on the set cs(g) of conjugacy class sizes of a finite group G. A vertex of a graph is called complete if it is adjacent to all the other vertices; as we will see, the condition that the prime graph on cs(g) has few complete vertices yields some significant restrictions on the group structure of G. Among other things, we can completely characterize the situation when this graph is non-complete and regular. Preliminary version June 6, 202 8:37

9 The Sabinin product for loops *P. Plaumann, L. Sabinina We give a description of L.Sabinin s construction ot the left multiplication group of a loop using its left inner mapping group (left transassociant). Preliminary version June 6, 202 4:40

10 THE ZARISKI AND MARKOV TOPOLOGIES ON A GROUP DANIELE TOLLER Given a group G, Markov [4] defined an elementary algebraic subset X of G to be the solution-set of a one-variable equation over G, i.e. X = {x G g x ε g 2 x ε2 g n x εn = e G } for n N, ε,..., ε n Z, and g,..., g n G. He also defined a subset X to be unconditionally closed in G if it were closed in every Hausdorff group topology on G. These definitions implicitly introduced two T topologies on G, now called the Zariski topology Z G of G, generated by the elementary algebraic subsets as closed sets, and the Markov topology M G of G, generated by the unconditionally closed subsets as closed sets (this topology is the intersection of every Hausdorff group topology on G, and it need not to be neither a group topology, nor even Hausdorff). One can easily see that Z G M G, and Markov himself proved that they coincide if the group G is countable. He asked whether these two topologies always coincide, and Hesse [5] built an example showing that this is not true in general. The authors of [2] and [6] indipendently proved that the equality Z G = M G holds for groups G = A ( i I G i), where A is an abelian group and each group Gi is countable. The more topologies the group G carries, the coarser M G is; while M G is itself a Hausdorff group topology if and only if it is contained in every Hausdorff group topology on G. In this case, M G is the minimum in the lattice of all Hausdorff group topologies on G. If this happens, the group G is called algebraically minimal. For example, it is a classic result [3] that, for the group S(X) of the permutations of an infinite set X, the point-wise convergence topology is contained in every Hausdorff group topology on S(X), so that S(X) is algebraically minimal. Let S ω (X) be the subgroup of S(X) consisting of the finite-support permutations. It has been very recently proved in [] that, for every subgroup G of S(X), containing S ω (X), Z G = M G is the point-wise convergence topology on G. This extends Gaughan s result, and in particular all such subgroups are algebraically minimal. References [] T. Banakh, I. Guran, I. Protasov: Algebraically determined topologies on permutation groups, Topology Appl. 59 (202), no. 9, pp [2] D. Dikranjan, D. Shakhmatov, Reflection principle characterizing groups in which unconditionally closed sets are algebraic, J. Group Theory (2008), no. 3, pp [3] E.D. Gaughan, Topological group structures of infinite symmetric groups, Proc. Nat. Acad. Sci. U.S.A. 58 (967), pp [4] A.A. Markov, On unconditionally closed sets, Comptes Rendus Doklady AN SSSR (N.S.) 44 (944), pp (in Russian). Date: Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, 3300 Udine, Italy. daniele.toller@uniud.it.

arxiv: v1 [math.gn] 24 Oct 2013

arxiv: v1 [math.gn] 24 Oct 2013 The Bridge Theorem for totally disconnected LCA groups arxiv:1310.6665v1 [math.gn] 24 Oct 2013 Dikran Dikranjan dikran.dikranjan@uniud.it Dipartimento di Matematica e Informatica, Università di Udine,