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Given a connected graph X and a group G of its automorphisms we first introduce an approach for constructing all pairwise nonequivalent connected solvable regular coverings ℘ : X ˜ → X (that is, with a solvable group of covering transformations CT ( ℘ ) ) along which G lifts, up to a prescribed order n of X ˜ . Next, given a connected solvable regular covering ℘ : X ˜ → X by means of voltages and a group G ≤ Aut ( X ) that lifts along ℘, we consider algorithms for testing whether the lifted group G ˜ is a split extension of CT ( ℘ ) . In computational group theory, methods for testing whether a given extension of permutation groups splits are known. However, in order to apply the existing algorithms, X ˜ together with CT ( ℘ ) and G ˜ need to be constructed in the first place, which is far from optimal. Recently, an algorithm avoiding such explicit constructions has been proposed by Malnic and Požar (submitted for publication) . We here provide additional details about this algorithm and investigate its performance compared to the one using explicit constructions. To this end, a concrete dataset of solvable regular covers of graphs has been generated by the algorithm mentioned in the first paragraph.

The dynamical system on T^2 which is a group extension over an irrational rotation on T^1 is investigated. The criterion when the extension is minimal, a system of order 2 and when the maximal equicontinuous factor is the irrational rotation is given. The topological complexity of the extension is computed, and a negative answer to the latter part of an open question raised by Host-Kra-Maass is obtained., 17 pages, accepted for publication in SCIENCE CHINA Mathematics

We identify the class of elementary groups: the smallest class of totally disconnected locally compact second countable (t.d.l.c.s.c.) groups that contains the profinite groups and the discrete groups, is closed under group extensions of profinite groups and discrete groups, and is closed under countable increasing unions. We show this class enjoys robust permanence properties. In particular, it is closed under group extension, taking closed subgroups, taking Hausdorff quotients, and inverse limits. A characterization of elementary groups in terms of well-founded descriptive-set-theoretic trees is then presented. We conclude with three applications. We first prove structure results for general t.d.l.c.s.c. groups. In particular, we show a compactly generated t.d.l.c.s.c. group decomposes into elementary groups and topologically characteristically simple groups via group extension. We then prove two local-to-global structure theorems: Locally solvable t.d.l.c.s.c. groups are elementary and [A]-regular t.d.l.c.s.c. groups are elementary., Accepted version. To appear in The Proceedings of the London Mathematical Society

The Benson-Solomon systems comprise the one currently known family of simple saturated fusion systems at the prime two that do not arise as the fusion system of any finite group. We determine the automorphism groups and the possible almost simple extensions of these systems and of their centric linking systems.

Given a group G and a G-module A, we show how to determine up to isomorphism the extensions E of A by G so that A embeds as smallest non-trivial term of the derived series or of the lower central series into E., 13 pages, 2 figures

We investigate the universal cover of a topological group that is not necessarily connected. Its existence as a topological group is governed by a Taylor cocycle, an obstruction in 3-cohomology. Alternatively, it always exists as a topological 2-group. The splitness of this 2-group is also governed by an obstruction in 3-cohomology, a Sinh cocycle. We give explicit formulas for both obstructions and show that they are inverse of each other., 9 pages. Version 2: historical review added. Version 3 (14 pages): historical review revised, minor corrections elsewhere, some of the results did not make it into the final version. Version 4: final journal version with a slightly different (to Version 3) approach

In this paper we extend and generalize the Calabi construction for flat manifolds to flat orbifolds. Let Γ be an n-dimensional crystallographic group (n-space group) that is Z -reducible. Then the flat orbifold E n / Γ has a nontrivial fibered orbifold structure. We prove that this structure can be described by a Calabi construction, that is, E n / Γ is represented as the quotient of the Cartesian product of two flat orbifolds under the diagonal action of a structure group of isometries. We determine the structure group and prove that it is finite if and only if the fibered orbifold structure has an orthogonally dual fibered orbifold structure. A fibered orbifold structure on E n / Γ corresponds to a space group extension 1 → N → Γ → Γ / N → 1 . We give a condition for the splitting of a space group extension in terms of the structure group action that is strong enough to detect the splitting of all the space group extensions corresponding to the standard Seifert fibrations of a compact, connected, flat 3-orbifold.

Let K be a normal subgroup of the finite group H. To a point of a K-interior H-algebra we associate a group extension, and we prove that this extension is isomorphic to an extension associated to the point given by the Brauer homomorphism. This may be regarded as a generalization and an alternative treatment of Dade's results [2, Section 12].

Using a recent result of Stadlbauer for group extensions of Holder continuous potentials on a countable Markov shift, we characterise amenability in terms of the spectral radius of the Perron-Frobenius operator associated to the group extended Markov system. This generalises a result of Day for random walks on groups. Moreover, we show that if the potential has symmetry with respect to the group, then the spectral radius is given by the Gurevipressure of the potential.

The work is inspired by an article of Herzog, Longobardi, and Maj, who considered groups with a finite number of infinite conjugacy classes. Their main results were obtained under assumption that the FC-center is of finite index in the group. We consider here infinite groups with a finite number of conjugacy classes of any size (FNCC-groups). Hence the FC-center in our case will be finite, but of infinite index in the group. Among results on these groups we give a criterion for a wreath product of FNCC-groups to be an FNCC-group.

The aim of this paper is to study co-prolongations of central extensions. We construct the obstruction theory for co-prolongations and classify the equivalence classes of these by kernels of a homomorphisms between 2-dimensional cohomology groups of groups., 9 pages

We define what it means to ‘speed up’ a-measure-preserving dynamical system, and prove that given any ergodic extension Tσ of a -measure-preserving action by a locally compact, second countable group G, and given any second G-extension Sσ of an aperiodic -measure-preserving action, there is a relative speedup of Tσ, which is relatively isomorphic to Sσ. Furthermore, we show that given any neighbourhood of the identity element of G, the aforementioned speedup can be constructed so that the transfer function associated with the isomorphism between the speedup and Sσ almost surely takes values only in that neighbourhood.

In this paper we introduce the notion of Boolean like ring extension(for short, BLR extension) of a group and obtain various properties. We show that any BLR extension of a group is a group. Further, scalar multiplication on any BLR extension of a group is defined and studied certain properties.

Covering techniques have recently emerged as an effective tool used for classification of several infinite families of connected symmetric graphs. One commonly encountered technique is based on the concept of lifting groups of automorphisms along regular covering projections ?$\wp \colon \tilde{X} \to X$?. Efficient computational methods are known for regular covers with cyclic or elementary abelian group of covering transformations CT?$(\wp)$?. In this paper we consider the lifting problem with an additional condition on how a group should lift: given a connected graph ?$X$? and a group ?$G$? of its automorphisms, find all connected regular covering projections ?$\wp \colon \tilde{X} \to X$? along which ?$G$? lifts as a sectional split extension. By this we mean that there exists a complement ?$\overline{G}$? of CT?$(\wp)$? within the lifted group ?$\tilde{G}$? such that ?$\overline{G}$? has an orbit intersecting each fibre in at most one vertex. As an application, all connected elementary abelian regular coverings of the complete graph ?$K_4$? along which a cyclic group of order 4 lifts as a sectional split extension are constructed. Krovne tehnike so se izkazale kot učinkovito orodje pri klasifikaciji več neskončnih družin povezanih simetričnih grafov. Ena izmed pogostih tehnik, s katerimi se srečamo, temelji na konceptu dviga avtomorfizmov grup vzdolž regularnih krovnih projekcij ?$\wp \colon \tilde{X} \to X$?. Učinkovite računske metode so znane v primeru regulanih krovov s ciklično ali elementarno abelsko grupo krovnih transformacij CT?$(\wp)$?. V članku študiramo problem dviga pri dodatnem pogoju, kako naj se grupa dvigne: za dani povezan graf? $X$? in podgrupo ?$G$? njegovih avtomorfizmov poišči vse povezane regularne krovne projekcije ?$\wp \colon \tilde{X} \to X$?, vzdolž katerih se ?$G$? dvigne kot sekcijska razcepna razširitev. To pomeni, da obstajakomplement ?$\overline{G}$? k CT?$(\wp)$? znotraj dvignjene grupe ?$\tilde{G}$?, tako da ima ?$\overline{G}$? orbito, ki seka vsako vlakno v največ enem vozlišču. Za ilustracijo konstruiramo vse povezane elementarno ableske regularne krove polnega grafa ?$K_4$?, vzdolž katerih se ciklična grupa reda 4 dvigne kot sekcijska razcepna razširitev.

Let G be a finite group with a normal subgroup N that has a p-block b of defect 0. As is well-known, we may assume that b is G-stable. Then we show that there is an extension of G/N by a central cyclic p′-subgroup and a p-block of such that the blocks of G covering b biject with the blocks of that cover and corresponding blocks have several equivalent properties. These results extend the known results in the situation that N is a normal p′-subgroup.

We describe unary polynomial functions on nite groups G that are semidirect products of an elementary abelian group of exponent p and a cyclic group of prime order q, p ≠ q.

There are investigated supergroups of some hyperbolic space groups with simplicial fundamental domain. Six simplices considered here from [9] are collected in families F9 (T23, T64), F10 (T21, T49, T61), F29 (T34). All of them have the same symmetry by half-turn h, with axis through the midpoints of edges A0A1 and A2A3. Since that isometry identifies pairs of points, if a supergroup with such smaller fundamental domain exists, it is of index 2. At the side pairings of T34 this half-turn implies additional reflections, equal parameters 2a = 6b, and leads to Family 2, considered in [9]. Other possibility to find supergroups is when the simplices have vertices out of the absolute. In that case we can truncate them by polar planes of the vertices and the new polyhedra are fundamental domains of richer groups.

The main results of this note extend a theorem of Kesten for symmetric random walks on discrete groups to group extensions of topological Markov chains. In contrast to the result in probability theory, there is a notable asymmetry in the assumptions on the base. That is, it turns out that, under very mild assumptions on the continuity and symmetry of the associated potential, amenability of the group implies that the Gurevic-pressures of the extension and the base coincide whereas the converse holds true if the potential is H\"older continuous and the topological Markov chain has big images and preimages. Finally, an application to periodic hyperbolic manifolds is given., Comment: New proof of Lemma 5.3 due to the gap in the first version of the article

An elliptic curve E defined over a finite field K, E(K) is the set of solutions to the general Weierstrass polynomial E: y 2 + a1xy + a3y = x 3 + a2x 2 + a4x + a6 where the coefficients a1, a2, a3, a4, a6 є K. There exist a well defined addition of points on each curve such that the points form an abelian group under the addition operation. This group is either cyclic or isomorphic to the product of two cyclic groups. These set of solutions that form the group lie in the closure of the field K over which the curve is defined. If we allow the set to lie only in a particular extension of K, the addition operation is well defined there too. Therefore we can associate a group to every extension K' of the field K denoted by E(K'). Will the structure of the group defined over the base field K, be affected if the same group is made to lie in the extension K' of K?

In this paper, we introduce a symmetric continuous cohomology of topological groups. This is obtained by topologizing a recent construction due to Staic (J. Algebra 322 (2009), 1360-1378), where a symmetric cohomology of abstract groups is constructed. We give a characterization of topological group extensions that correspond to elements of the second symmetric continuous cohomology. We also show that the symmetric continuous cohomology of a profinite group with coefficients in a discrete module is equal to the direct limit of the symmetric cohomology of finite groups. In the end, we also define symmetric smooth cohomology of Lie groups and prove similar results., Comment: 23 pages, to appear in Homology Homotopy and Applications

When the benefit of making a correct decision is sufficiently high, even a slight increase in the probability of making such a decision justifies an increase in the number of decision makers. Applying a standard uncertain dichotomous choice benchmark setting, this study focuses on the relative desirability of two alternatives: adding individuals with capabilities identical to the existing ones and adding identical individuals with mean-preserving capabilities that depend on the states of nature. Our main result establishes that when the group applies the simple majority rule, variability in the capabilities of the new decision makers under the two states of nature, which is commonly observed in various decision-making settings, is less desirable in terms of the probability of making the correct decision.

In this paper, we construct a seven-term exact sequence involving the cohomology groups of a group extension. Although the existence of such a sequence can be derived using spectral sequence arguments, there is little knowledge about some of the maps occuring in the sequence, limiting its usefulness. Here we present a construction using only very elementary tools, always related to the notion of conjugation in a group. This results in a complete and usable description of all the maps, which we describe both on cocycle level as on the level of the interpretations of low dimensional cohomology groups (e.g. group extensions)., Comment: Updated version, new cocycle description added

We compute the topological K-theory of the group C*-algebra [Formula: see text] for a group extension 1 → ℤn → Γ → ℤ/m → 1 provided that the conjugation action of ℤ/m on ℤn is free outside the origin.

We prove that for all ergodic extensions S1 of a transformation by a locally compact second countable group G, and for all G extensions S2 of an aperiodic transformation, there is a relative speedup of S1 that is rela- tively isomorphic to S2. We apply this result to give necessary and sufficient conditions for two ergodic n point or countable extensions to be related in this way. Let (X,B,µ) be a Lebesgue probability space and T : X → X an ergodic µ−preserving automorphism. By a speedup of T we mean an automorphism of X of the form x 7→T p(x) (x), where p is a positive integer-valued function on X. We denote such an automorphism by T p . It is natural to ask which automorphisms (up to isomorphism) can be obtained from T in this way. If p is integrable there are significant restrictions on the possible speedups of T. It was proved in (N), for example, that if S is isomorphic to T p(x) and R pdµ < ∞, then the entropies of S and T satisfy h(S) = R pdµ � h(T). As another example, from (OW) we see that if S is isomorphic to T p(x) and R pdµ < ∞, then T is a factor of a finite measure preserving transformation that induces S. Thus, if S is loosely Bernoulli, it can only be expressed as an integrable speedup of T if T is also loosely Bernoulli. However, if p is not required to be integrable, then there are no obstructions to this relation; in (AOW) the general result was proved that for all ergodic finite measure preserving automorphisms T and all aperiodic finite measure preserving S, there is a speedup of T that is isomorphic to S. In this paper we prove a conditional version of that result in the case of ergodic group extensions, and we give an application of this result to the classification of ergodic finite extensions. Suppose that (X,B,µ) and T are as above, and T is a factor of an automorphism S of the space (Y,C,�). Then by a speedup of S relative to T we mean a speedup S p where p is measurable with respect to the factor (X,B,µ). Of particular interest to us it the case where S is a group extension of T by a locally compact second countable group G. We recall some basic definitions: Let G be a locally compact second countable group with left Haar measure �, and let � : X ×Z → G be a cocycle for T. That is, � is a measurable function such that for almost all x ∈ X and all n,m ∈ Z

We state the notion of an extension datum of a root system. This concept generalizes the root system extended by an abelian group. We classify them, and show that there is a one-to-one correspondence between (reduced) root systems extended by an abelian group and (reduced) tame irreducible extension data. We also show that an extension datum, modulo some isotropic roots, is the union of a finite number of tame irreducible extension data.

Cohomology groups H s ( Z n , Z m ) are studied to describe all groups up to isomorphism which are (central) extensions of the cyclic group Z n by the Z n -module Z m . Further, for each such a group the number of non-equivalent extensions is determined.

Suppose G is either a soluble (torsion-free)-by-finite group of finite rank or a soluble linear group over a finite extension field of the rational numbers. We consider the implications for G if G has an automorphism of finite order m with only finitely many fixed points. For example, if m is prime then G is a finite extension of a nilpotent group and if m = 4 then G is a finite extension of a centre-by-metabelian group. This extends the special cases where G is polycyclic, proved recently by Endimioni (2010); see [3] .

Given a principal bundle G hooked right arrow P -> B (each being compact, connected and oriented) and a G-invariant metric h(P) on P which induces a volume form mu(P), we consider the group of all unimodular automorphisms SAut(P, mu(P)) := {phi is an element of Diff(P) vertical bar phi*mu(P) = mu(P) and phi is G-equivariant) of P, and determines its Euler equation a la Arnold. The resulting equations turn out to be (a particular case of) the Euler-Yang-Mills equations of an incompressible classical charged ideal fluid moving on B. It is also shown that the group SAut(P, mu(P)) is an extension of a certain volume preserving diffeomorphisms group of B by the gauge group Gau(P) of P. (C) 2010 Elsevier B.V. All rights reserved.

Let H be a Hopf algebra, and A an H-Galois extension. We investigate H-Morita autoequivalences of A, introduce the concept of H-Picard group, and we establish an exact sequence linking the H-Picard group of A and the Picard group of AcoH.

Some extension theorems for l-group-valued M-measures are proved.

We prove that a sequence of marked three-generated groups isomorphic to the iterated monodromy group of z 2 + i converges to a group of non-uniform exponential growth, which is an extension of the infinite direct sum of cyclic groups of order 4 by a Grigorchuk group.

In order to better understand the structure of indecomposable projective Mackey functors, we study extension groups of degree 1 between simple Mackey functors. We explicitly determine these groups between simple functors indexed by distinct normal subgroups. We next study the conditions under which it is possible to restrict ourselves to that case, and we give methods for calculating extension groups between simple Mackey functors which are not indexed by normal subgroups. We then focus on the case where the simple Mackey functors are indexed by the same subgroup. In this case, the corresponding extension group can be embedded in an extension group between modules over a group algebra, and we describe the image of this embedding. In particular, we determine all extension groups between simple Mackey functors for a p-group and for a group that has a normal p-Sylow subgroup. Finally, we compute higher extension groups between simple Mackey functors for a group that has a p-Sylow subgroup of order p.

We use our criterion for minimality of restrictions of compact minimal abelian flows to characterize minimality of group-extensions, products and restrictions of suspensions in terms of eigenvalues. In particular, we get a new proof of a theorem of Parry about minimality of group-extensions and generalize some theorems about minimality of products and restrictions of suspensions.