Fusions of Character Tables II: p-Groups

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.

In this chapter, we discuss several special types of modules that will make frequent appearances later. These include semisimple modules, artinian modules, and torsionfree modules, which have prototypes in vector spaces, finite abelian groups, and torsionfree abelian groups, respectively. That special types of modules have useful roles in the study of arbitrary modules may be seen already in the case of abelian groups (i.e., ℤ-modules). In studying an arbitrary abelian group, an almost reflexive first step is to look at its torsion part (i.e., the torsion subgroup) and its torsionfree part (i.e., the factor group modulo the torsion subgroup), since entirely different techniques are available (and needed) for dealing with torsion groups and torsionfree groups. On the torsionfree side, vector spaces make an appearance due to the facts that the torsionfree divisible abelian groups are exactly the vector spaces over ℚ and that every torsionfree abelian group can be embedded in a divisible one. On the torsion side, many questions can be reduced to the case of finite abelian groups, since every torsion abelian group is a directed union of finite subgroups. In studying torsion abelian groups, one also reduces to the case of p -groups for various primes p . Vector spaces make another appearance here, since in an abelian p -group the set of elements of order p (together with 0) forms a vector space over ℤ/ p ℤ.

Let an element of a group be called an FC element if it has only a finite number of conjugates in the group. Baer(1) and Neumann (8) have discussed groups in which every element is FC, and called them FC-groups. Both Abelian and finite groups are trivially FC-groups; Neumann has studied the properties common to FC-groups and Abelian groups, and Baer the properties common to FC-groups and finite groups. Baer has also shown that, for an arbitrary group G, the set H1 of all FC elements is a characteristic subgroup. Haimo (3) has defined the FC-chain of a group G byHi/Hi−1 is the subgroup of all FC elements in G/Hi−1.

Maximally localized Gabor orthonormal bases on locally compact Abelian groups

Abelian groups is the most famous in group theory and is also one of the most important groups, the study of abelian groups occupies a very important position in the group theory, abelian groups for almost all the research plays a backbone role, this article through replace new notation to the abelian groups of the term for a group, introduce some new definitions such as p-primary group, p-primary components of G, independent of groups and the Sylow subgroups for tool are discussed, and finally got about finite abelian groups of several important conclusions.

Let G G be a finite group and let f : X → Y f : X \to Y be a degree 1, G G -framed map such that X X and Y Y are simply connected, closed, oriented, smooth manifolds of dimension n = 2 k ≧ 6 n = 2k \geqq 6 and such that the dimension of the singular set of the G G -space X X is at most k k . In the previous article, assuming f f is k k -connected, we defined the G G -equivariant surgery obstruction σ ( f ) \sigma (f) in a certain abelian group. There it was shown that if σ ( f ) = 0 \sigma (f) = 0 then f f is G G -framed cobordant to a homotopy equivalence f ′ : X ′ → Y f’ : X’ \to Y . In the present article, we prove that the obstruction σ ( f ) \sigma (f) is a G G -framed cobordism invariant. Consequently, the G G -surgery obstruction σ ( f ) \sigma (f) is uniquely associated to f : X → Y f : X \to Y above even if it is not k k -connected.

Recently, the concept of (normal) bent partitions, which are partitions of elementary abelian groups having similar properties to spreads, has been introduced by Anbar and Meidl. A large number of bent partitions, so-called generalized semifield spreads, can be obtained from semifields with certain properties. A strongly related concept, namely Latin square partial difference set packings (LP-packings) in finite abelian groups, has also been introduced recently by Jedwab and Li. The examples for LP-packings in an elementary abelian group are obtained from spreads. LP-packings yield bent partitions (not only for elementary abelian groups). In this paper, we first point out that conversely, generalized semifield spreads yield LP-packings. As a result, there is a huge amount of LP-packings in elementary abelian groups, other than spreads. With some examples from ternary bent functions, we then show that normal bent partitions and LP-packings are not the same concept. Finally, we extend the lifting procedure from spreads to LP-packings in nonelementary abelian groups to a lifting procedure from some generalized spreads to LP-packings in nonelementary abelian groups and in larger elementary abelian groups. This potentially yields bent partitions other than generalized semifield spreads.

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

Brouéʼs abelian defect group conjecture holds for the sporadic simple Conway group [formula omitted

Let \Lambda be a finite abelian group. A dynamical system with transformation group \Lambda is a triple (A,\Lambda,\alpha) , consisting of a unital locally convex algebra A , the finite abelian group \Lambda and a group homomorphism \alpha:\Lambda\rightarrow\operatorname{Aut}(A) , which induces an action of \Lambda on A . In this paper we present a new, geometrically oriented approach to the noncommutative geometry of principal bundles with finite abelian structure group based on such dynamical systems.

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.

In this paper, we shall study semi-free finite group actions on homotopy spheres. In [8], M. Sebastiani studied semi-free finite group actions on homotopy spheres with two points as the fixed point set. He showed that the collection of equivariant diffeomorphism classes of these semi-free finite group actions is an abelian group under the equivariant connected sum about a fixed point. By the methods analogous to Kervaire-Milnor [4], he proved that this abelian group is a finite group in the case of cyclic group actions.

A BV-algebra structure on Hochschild cohomology of the integral group ring of finitely generated Abelian groups

Let A be an abelian group. If for any we define , then is a quandle called the Takasaki quandle and is denoted by T(A). In this article, we focus on the lattice of subquandles of a finite Takasaki quandle T(A), denoted by . Indeed, we characterize the subquandles of T(A) and show that is the lattice of cosets of A if and only if A is an odd abelian group or its 2-Sylow subgroup is a cyclic group. Further, we determine the homotopy type of the lattice of subquandles of T(A) which is the wedge of r spheres of dimension d, where r is the number of prime numbers dividing and d is determined by applying the fundamental theorem of finite abelian groups. Furthermore, we prove that is graded if and only if A is an odd abelian group or its 2-Sylow subgroup is cyclic or its 2-Sylow subgroup is elementary abelian. Moreover, we show that the lattice of subquandles of a finite Takasaki quandle is determined by a unique Takasaki quandle. In other words, we show that the lattices of subquandles of two finite Takasaki quandles T(A) and T(B) are isomorphic if and only if A and B are isomorphic. However, the lattice of subquandles of a quandle which is not a Takasaki quandle can be isomorphic to the lattice of subquandles of a Takasaki quandle. We show this by providing a quandle Q of order 8 which is not a Takasaki quandle, but is isomorphic to

In this paper we present a generalization of the main aspects of the basis theorem for pairs (G,H) in which H is a subgroup of a finite abelian group G, see [4], to the case in which G is an abelian group of bounded order. The proof we outline here is actually simpler than the one sketched in [4]. We remark that the assumption of the well ordering principle which occurs in the statement of the theorem is not an essential feature of it.KeywordsAbelian GroupInduction HypothesisBasis ElementDirect SummandBasis TheoremThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.