Solutions of the nth-order Cauchy difference equation on groups

This paper is concerned with the homotopy theory and simple homotopy theory of connected finite 2 dimensional CW complexes with finite cyclic fundamental group. The main theorem presents a complete classification of such complexes up to homotopy type, and this theorem has the corollary that homotopy type and simple homotopy type coincide for these complexes. This work is motivated by the general problem of describing the sets HT(n) and SHT(n) of homotopy types and simple homotopy types of n-complexes, that is, connected finite 2 dimensional CW-complexes with a given fundamental group n. A visually satisfying description of HT (n) or SHT (n) is that of either set as a graph whose edges connect the type of each n-complex X to the type of its sum X v S 2 with the 2-sphere S 2. These graphs are actually trees; they clearly contain no circuits, and they are connected because any two n-complexes have the same type once each is summed with an appropriate number of copies of the 2-sphere S 2. To re establish this latter observation ofJ. H. C. Whitehead ([20, Theorem 12]), note that each n-complex has the simple homotopy type of one modeled in an obvious fashion on some finite presentation of the fundamental group ~z (see Proposition 1). But two finite presentations of the same group n differ by a finite sequence of Tietze operations, two of which leave the simple homotopy type of the associated topological model unchanged, while two alter the simple homotopy type by an S 2 summand. Of special interest in each of these trees are the roots and the junctions. The roots are the (simple) homotopy types that do not admit a factorization involving an S 2 summand; they generate the rest of the types in the tree under the operation of forming sum with S 2. The junctions are the (simple) homotopy types that admit two or more inequivalent factorizations involving an S 2 summand; they determine the shape of the tree. Each junction is a 2-dimensional instance of non-cancellation of the 2-sphere S 2 with respect to the sum operation. When the group n is a free group F of finite rank or is the finite cyclic group Z~ of prime order q, complete descriptions of the trees HT (n) and SHT (n) can be derived from the literature, as follows. A result of C. T. C. Wall ([17, Proposition 3.3]) can be specialized to read that for a free group F of finite rank r every F-complex has the homotopy type of a sum of r

Let \({(S,\cdot)}\) be a semigroup, (H, +) an abelian group and \({f: S \to H}\). The first and second order Cauchy differences of f are $$\begin{aligned} C^{1}f(x, y)= & {} f(xy) - f(x) - f(y),\\ C^{2}f(x, y, z)= & {} f(xyz) - f(xy) - f(yz) - f(xz) + f(x) + f(y) + f(z).\end{aligned}$$ Higher order Cauchy differences Ckf are defined recursively. In the case of H = R, a ring where multiplication is distributive over addition, we show that functions \({f: S\to R}\) with vanishing finite Cauchy differences are closed under multiplication. The equation Ckf = 0 is considered for cyclic groups, free abelian groups and other selected quotients of the free groups.

The classifying space BG of a topological group G can be filtered by a sequence of subspaces B(q,G) using the descending central series of free groups. If G is finite, describing them as homotopy colimits is convenient when applying homotopy theoretic methods. In this thesis we introduce natural subspaces B(q,G)_p of B(q,G) defined for a fixed prime p. We show that B(q,G) is stably homotopy equivalent to a wedge sum of B(q,G)_p as p runs over the primes dividing the order of G. Colimits of abelian groups play an important role in understanding the homotopy type of these spaces. Extraspecial p-groups are key examples, for which these colimits turn out to be finite. We prove that for extraspecial p-groups of rank at least 4 the space B(2,G) does not have the homotopy type of a K(π,1) space. Furthermore, we give a group theoretic condition, applicable to symmetric groups and general linear groups, which implies the space B(2,G) not having the homotopy type of a K(π,1) space. For a finite group G, we compute the complex K-theory of B(2,G) modulo torsion.

Results and questions on matchings in abelian groups and vector subspaces of fields

We recall that the two-sided crossed product of finite cyclic groups is actually a generalization of the crossed product construction of the same type of groups (cf. [10]). In this paper, by considering the crossed and two-sided crossed products obtained from both finite and infinite cyclic groups, we first present the complete rewriting systems and normal forms of elements over crossed products. (We should note that the complete rewriting systems and normal forms of elements over two-sided crossed products have been recently defined in [10]). In the crossed product case, we will consider their presentations that were given in [2]. As a next step, by using the normal forms of elements of these two products, we calculate the growth series of the crossed product of different combinations of finite and infinite cyclic groups as well as the growth series of two-sided crossed product of finite cyclic groups.

We show that the class of groups which have monoid presentations by means of finite special [λ]-confluent string-rewriting systems strictly contains the class of plain groups (the groups which are free products of a finitely generated free group and finitely many finite groups), and that any group which has an infinite cyclic central subgroup can be presented by such a string-rewriting system if and only if it is the direct product of an infinite cyclic group and a finite cyclic group.

In this chapter we develop the algebra underlying multirate filter structures over free abelian groups of finite rank. Suppose A is a free abelian group of rank N. Each subgroup Δ of A is a free abelian group of rank n ≤ N. The basis theorem describes how Δ sits in A in the sense that it characterizes the quotient group A/Δ. This quotient group can be a finite abelian group a free abelian group of finite rank the direct product of a finite abelian group and a free abelian group of finite rank.

This thesis presents some contributions in additive combinatorics and arithmetic Ramsey theory. More specifically, it deals with the interaction between combinatorics, number theory and additive combinatorics. This area saw a great improvement with the Szemerédi Regularity Lemma and some of the results that followed. The Regularity Lemma and its consequences have become a widely used tool in graph theory, combinatorics and number theory. Furthermore, its language and point of view has deeply changed the face of additive number theory, a fact universally acknowledged by the Abel award given to Szemerédi in 2012. One of the main reasons for the prize has been Szemerédi's theorem, a result regarding the existence of arbitrarily long arithmetic progressions in dense sets of the integers, the proof of which uses the Regularity Lemma in a key step. One of the earlier consequences of the Regularity Lemma was the Removal Lemma for graphs that was used by Ruzsa and Szemerédi to show Roth theorem, regarding the existence of 3-term arithmetic progressions in dense sets of the integers, in a combinatorial way. The Removal Lemma states that in any graph K with few copies of a subgraph, say a triangle, we can remove few edges from K so that the result contains no copy of the subgraph. This has become a key tool in the applications of the so-called Regularity Method, which has extensive literature in combinatorics, graph theory, number theory and computer science. In 2005 Green introduced a regularity lemma for Abelian groups as well as an algebraic removal lemma. The removal lemma for groups states that, for a given finite Abelian group G, if there are o(|G|^3) solution to x+y+z+t=0 with the variables taking values in S, a subset of G, then we can remove o(|G|) elements from S to make the set S solution-free. The main contributions of this work corresponds to extensions of the removal lemma for groups to either more general contexts, like non-necessary Abelian finite groups, or to linear systems of equations for finite Abelian groups. The main goal is to give a comprehensive and more general framework for many results in additive number theory like Szemerédi Theorem. In particular, we show that the removal lemma for groups by Green can be extended to non-necessary Abelian finite groups. Moreover, we prove a removal lemma for linear systems on finite fields: for every e>0 there exists a d>0 such that if A is a (k x m) linear system of equations with coefficients in a finite field F and the number of solutions to Ax=b, where each variable takes values from a subset Si in F is less than d times |F| raised to m-k, then by removing less than e|F| elements in each Si we can make the resulting sets solution-free, thus solving a conjecture by Green to that respect. Even more, if A is an integer linear system, G is a finite Abelian group, and the determinantal of A and |G| are coprime, then a similar statement holds. Let us mention that the last result allows us to characterize those linear systems where any set S with size proportional to G has a nontrivial solution in S, provided |G| is large enough. This extends the validity of Szmerédi's theorem to finite Abelian groups. These extensions of the removal lemma have been used in arithmetic Ramsey theory to obtain counting results for the number of monochromatic solutions of linear systems. The main result from a work by Frankl, Graham and Rödl in '88 states that the number of monochromatic solutions of regular systems in integer intervals is in fact a positive proportion of the total number of solutions. We give analogous results for solutions in Abelian groups with bounded exponent, for which the main tool in the torsion-free case cannot be applied. Density versions of these counting results are also obtained, in this case with a full characterization.

A skew morphism of a finite group B is a permutation φ of B that preserves the identity element of B and has the property that for every a ∈ B there exists a positive integer ia such that φ(ab) = φ(a)φia(b) for all b ∈ B. The problem of classifying skew morphisms for all finite cyclic groups is notoriously hard, with no such classification available up to date. Each skew morphism φ of Zn is closely related to a specific skew morphism of Z|⟨φ⟩|, called the quotient of φ. In this paper, we use this relationship and other observations to prove new theorems about skew morphisms of finite cyclic groups. In particular, we classify skew morphisms for all cyclic groups of order 2em with e ∈ {0, 1, 2, 3, 4} and m odd and square-free. We also develop an algorithm for finding skew morphisms of cyclic groups, and implement this algorithm in MAGMA to obtain a census of all skew morphisms for cyclic groups of order up to 161. During the preparation of this paper we noticed a few flaws in Section 5 of the paper Cyclic complements and skew morphisms of groups [J. Algebra 453 (2016), 68–100]. We propose and prove weaker versions of the problematic original assertions (namely Lemma 5.3(b), Theorem 5.6 and Corollary 5.7), and show that our modifications can be used to fix all consequent proofs (in the aforementioned paper) that use at least one of those problematic assertions.

In this paper we find the general product induced by the semi special permutation π = (1)(2)(3)(4)(5)(6)(7). That is the general products of two finite cyclic groups in which one of order 7 and the other is of order m these general products can be described in terms of numerical parameters.

For any commutative semigroup S and positive integer m the power function f: S rightarrow S defined by f(x) = x^m is an endomorphism of S. We partly solve the Lesokhin–Oman problem of characterizing the commutative semigroups whose all endomorphisms are power functions. Namely, we prove that every endomorphism of a commutative monoid S is a power function if and only if S is a finite cyclic group, and that every endomorphism of a commutative ACCP-semigroup S with an idempotent is a power function if and only if S is a finite cyclic semigroup. Furthermore, we prove that every endomorphism of a nontrivial commutative atomic monoid S with 0, preserving 0 and 1, is a power function if and only if either S is a finite cyclic group with zero adjoined or S is a cyclic nilsemigroup with identity adjoined. We also prove that every endomorphism of a 2-generated commutative semigroup S without idempotents is a power function if and only if S is a subsemigroup of the infinite cyclic semigroup.

Long colimits of topological groups II: Free groups and vector spaces

On monodromy representations in Denham–Suciu fibrations

Having reduced the general theory of the FT on finite Abelian groups to the theory of the FT on finite cyclic groups, it suffices to study the FT of functions defined on the cyclic group Zn for an arbitrary value of n, where n > 1. Our next goals are the following: Determine the form of eigenvectors of the FT (Section 7.2). Find the spectral decomposition of the FT on Zn or, equivalently, the decomposition of the space VZn as a direct sum of its invariant subspaces (Section 7.3). Determine the multiplicity of the eigenvalues of the FT (Section 7.5).

Affine Weyl groups and their parabolic quotients are used extensively as indexing sets for objects in combinatorics, representation theory, algebraic geometry, and number theory. Moreover, in the classical Lie types we can conveniently realize the elements of these quotients via intuitive geometric and combinatorial models such as abaci, alcoves, coroot lattice points, core partitions, and bounded partitions. In [2], Berg et al. described a bijection between $$n$$n-cores with first part equal to $$k$$k and $$(n-1)$$(n-1)-cores with first part less than or equal to $$k$$k, and they interpret this bijection in terms of these other combinatorial models for the quotient of the affine symmetric group by the finite symmetric group. In this paper, we discuss how to generalize the bijection of Berg---Jones---Vazirani to parabolic quotients of affine Weyl groups in type $$C$$C. We develop techniques using the associated affine hyperplane arrangement to interpret this bijection geometrically as a projection of alcoves onto the hyperplane containing their coroot lattice points. We are thereby able to analyze this bijective projection in the language of various additional combinatorial models developed by Hanusa and Jones [10], such as abaci, core partitions, and canonical reduced expressions in the Coxeter group.