How often can a finite group be realized as a Galois group over a field?

Let G be any finite group and any class of fields. By we denote the minimal number of realizations of G as a Galois group over some field from the class . For G abelian and the class of algebraic extensions of ℚ we give an explicit formula for . Similarly we treat the case of an abelian p-group G and the class which is conjectured to be the class of all fields of characteristic ≠p for which the Galois group of the maximal p-extension is finitely generated. For non-abelian groups G we offer a variety of sporadic results. Received: 27 October 1998 / Revised version: 3 February 1999     

When is the Isbell topology a group topology?

Conditions on a topological space X under which the space C(X,R) of continuous real-valued maps with the Isbell topology κ is a topological group (topological vector space) are investigated. It is proved that the addition is jointly continuous at the zero function in Cκ(X,R) if and only if X is infraconsonant. This property is (formally) weaker than consonance, which implies that the Isbell and the compact-open topologies coincide. It is shown the translations are continuous in Cκ(X,R) if and only if the Isbell topology coincides with the fine Isbell topology. It is proved that these topologies coincide if X is prime (that is, with at most one non-isolated point), but do not even for some sums of two consonant prime spaces.     

When is a right orderable group locally indicable?

If a group has an ascending series of subgroups such that for each ordinal , and has no non-abelian free subsemigroup, then is right orderable if and only if it is locally indicable. In particular if is a radical-by-periodic group, then it is right orderable if and only if it is locally indicable.

     

When are small subgraphs of a random graph normally distributed?

Summary LetG be a graph and letX _n count copies ofG in a random graphK(n,p). The random variable is asymptotically normally distributed if and only ifnp ^m andn ² (1-p), wherem=max {e(H)/|H|:HG}. In addition to, and in connection with this main result we investigate the formula for Var (X _n ) and the Poisson convergence ofX _n .Part of the research was done during the author's stay in Division of Mathematics and Science, St. John's University, Staten Island, New York     

When does the top homology of a random simplicial complex vanish?

Several years ago Linial and Meshulam (Combinatorica 26 (2006) 457–487) introduced a model called of random n‐vertex d‐dimensional simplicial complexes. The following question suggests itself very naturally: What is the threshold probability at which the d‐dimensional homology of such a random d‐complex is, almost surely, nonzero? Here we derive an upper bound on this threshold. Computer experiments that we have conducted suggest that this bound may coincide with the actual threshold, but this remains an open question. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 26–35, 2015     

When is a group action determined by its orbit structure?

We present a simple approach to questions of topological orbit equivalence for actions of countable groups. For example, for any action of a countable group on a topological manifold where the fixed sets for any element are contained in codimension two submanifolds, every orbit equivalence is equivariant. Even in the presence of larger fixed sets, for actions preserving rigid geometric structures our results force sufficiently smooth orbit equivalences to be equivariant. For instance, if a countable group acts on and the action is C¹ orbit equivalent to the standard action of on , then is isomorphic to and the actions are isomorphic. (The same result holds if we replace by a finite index subgroup.) We also show that preserving a geometric structure is an invariant of smooth orbit equivalence and give an application of our ideas to the theory of hyperbolic groups.In the course of proving our theorems, we generalize a theorem of Sierpinski which says that a connected Hausdorff compact topological space is not the disjoint union of countably many closed sets. We prove a stronger statement that allows small intersections provided the space is locally connected. This implies that for any continuous action of a countable group on a connected, locally connected, locally compact, Hausdorff topological space, where the fixed set of every element is small, every orbit equivalence is equivariant.     

How do random Fibonacci sequences grow?

We study the random Fibonacci sequences defined by ${F_1 = F_2 = \widetilde F_1 = \widetilde F_2 = 1}$ and for n ≥ 1, F _n+2 = F _n+1 ± F _n (linear case) and ${\widetilde F_{n+2} = |\widetilde F_{n+1} \pm \widetilde F_{n}|}$ (non-linear case), where each ± sign is independent and either + with probability p or ? with probability 1 ? p (0 < p ≤ 1). Our main result is that the exponential growth of F _n for 0 < p ≤ 1, and of ${\widetilde F_{n}}$ for 1/3 ≤ p ≤ 1 is almost surely given by $${\int \limits _0^\infty \log x\, d\nu_\alpha (x),}$$ where α is an explicit function of p depending on the case we consider, and ν _α is an explicit probability distribution on ${\mathbb{R}_+}$ defined inductively on Stern–Brocot intervals. In the non-linear case, the largest Lyapunov exponent is not an analytic function of p, since we prove that it is equal to zero for 0 < p ≤ 1/3. We also give some results about the variations of the largest Lyapunov exponent, and provide a formula for its derivative.     

On the T∘-slices of a finite group

A slice $(G,S)$ of finite groups is a pair consisting of a finite group G and a subgroup S of G. In this paper, we show that some properties of finite groups extend to slices of finite groups. In particular, by analogy with B-groups, we introduce the notion of $T^{°}$-slice, and show that any slice of finite groups admits a largest quotient $T^{°}$-slice.     

The dimension of the finite subsets κ

We answer a question of M. Pouzet by showing that the Dushnik-Miller dimension of the finite subsets of the infinite cardinal ordered by inclusion is ().This paper was written when the first author was visiting the University of Calgary, June, 1995. Research partially supported by Office of Naval Research grant NOOO14-90-1206.Research supported by NSERC grant #69-0982.     

When do the Recession Cones of a Polyhedral Complex Form a Fan?

We study the problem of when the collection of the recession cones of a polyhedral complex also forms a complex. We exhibit an example showing that this is no always the case. We also show that if the support of the given polyhedral complex satisfies a Minkowski–Weyl-type condition, then the answer is positive. As a consequence, we obtain a classification theorem for proper toric schemes over a discrete valuation ring in terms of complete strongly convex rational polyhedral complexes.     

When are all group codes of a noncommutative group Abelian (a computational approach)?

Let G be a finite group and F be a field. Any linear code over F that is permutation equivalent to some code defined by an ideal of the group ring FG will be called a G-code. The theory of these ??abstract?? group codes was developed in 2009. A code is called Abelian if it is an A-code for some Abelian group A. Some conditions were given that all G-codes for some group G are Abelian but no examples of non-Abelian group codes were known at that time. We use a computer algebra system GAP to show that all G-codes over any field are Abelian if |G|?<?128 and |G| ? {24, 48, 54, 60, 64, 72, 96, 108, 120}, but for F?=? $ {\mathbb{F}_5} $ and G?=?S₄ there exist non-Abelian G-codes over F. It is also shown that the existence of left non-Abelian group codes for a given group depends in general on the field of coefficients, while for (two-sided) group codes the corresponding question remains open.     

On the Chermak–Delgado lattice of a finite group

Abstract

By imposing conditions upon the index of a self-centralizing subgroup of a group, and upon the index of the center of the group, we are able to classify the Chermak-Delgado lattice of the group. This is our main result. We use this result to classify the Chermak-Delgado lattices of dicyclic groups and of metabelian p-groups of maximal class.     

The asymptotic σ-algebra of a recurrent random walk on a locally compact group

Let μ be a probability measure on a locally compact second countable groupG defining a recurrent (but not necessarily Harris) random walk. Denote byG ^∞ the space of paths and byB ^(a)the asymptotic σ-algebra. Let the starting measure be equivalent to the Haar measure and writeQ for the corresponding Markov measure onG ^∞. We prove thatL ^∞(G^∞, B^(a), Q) is in a canonical way isomorphic toL ^∞(G/N) whereN is the smallest closed normal subgroup ofG such that μ(zN)=1 for somez∈G. The groupG/N is either a finite cyclic group with generatorzN or a compact abelian group having the cyclic group as a dense subgroup. As a corollary we obtain that the set of all φ∈L ¹(G) such that coincides with the kernel of the canonical mapping ofL ¹(G)ontoL ¹(G/N). In particular, when μ is aperiodic, i.e.,G=N, then the random walk is mixing: for every φ∈L ¹(G) with ∝ φ=0.     

When is a -adic power series an endomorphism of a formal group?

If is a noninvertible endomorphism of a formal group, then we have that commutes with an invertible series and is Galois over for all . We shall prove that the converse of this statement is also true.

     

When is a Coalgebra a Generator?

Let C be a coalgebra over a field k. The aim of this paper is to study the following problem : (P) If C is a k-coalgebra such that C is a generator for the category of left comodules, is C a left quasi-co-Frobenius coalgebra ? The converse always holds. We show that if C has a finite coradical series, the answer is positive.