Finite automata presentable abelian groups

We give new examples of FA presentable torsion-free abelian groups. Namely, for every n2, we construct a rank n indecomposable torsion-free abelian group which has an FA presentation. We also construct an FA presentation of the group in which every nontrivial cyclic subgroup is not FA recognizable.     

Finite quantum groups over abelian groups of prime exponent

We classify pointed finite-dimensional complex Hopf algebras whose group of group-like elements is abelian of prime exponent p, p>17. The Hopf algebras we find are members of a general family of pointed Hopf algebras we construct from Dynkin diagrams. As special cases of our construction we obtain all the Frobenius-Lusztig kernels of semisimple Lie algebras and their parabolic subalgebras. An important step in the classification result is to show that all these Hopf algebras are generated by group-like and skew-primitive elements.     

Finite groups with abelian second maximal subgroups

Abstract

In this article, we give a complete classification of finite groups whose second maximal subgroups are all abelian.     

Finite permutation groups containing a transitive abelian subgroup

In this paper, we classify finite permutation groups with a transitive abelian subgroup that are almost simple, quasiprimitive and innately transitive, which extend the results of Li and Praeger that is on finite permutation groups with a transitive cyclic subgroup.     

非极大交换子群皆正规的有限群

设H是有限群G的一个交换子群.如果H在G中的中心化子正是它本身,则称H为G的极大交换子群.本文主要研究每一非极大交换子群都正规的有限群的结构,对这类有限群给出其完全分类.     

Almost-affable abelian groups

In this paper we show that a direct summand of a simply presented mixed abelain group is an almost affable group. As a consequence, the classification theorem due to the author is extended to the largest possible class.     

Universal abelian groups

We examine the existence of universal elements in classes of infinite abelian groups. The main method is using group invariants which are defined relative to club guessing sequences. We prove, for example:Theorem:For n≧2, there is a purely universal separable p-group in ℵ _n if, and only if, . Partially supported by the United States-Israel Binational Science Foundation. Publication number 455.     

Graphically abelian groups

We define a group G to be graphically abelian if the function g?g⁻¹ induces an automorphism of every Cayley graph of G. We give equivalent characterizations of graphically abelian groups, note features of the adjacency matrices for Cayley graphs of graphically abelian groups, and show that a non-abelian group G is graphically abelian if and only if G=E×Q, where E is an elementary abelian 2-group and Q is a quaternion group.     

Quasi-minimal abelian groups

An abelian group is said to be quasi-minimal (purely quasi-minimal, directly quasi-minimal) if it is isomorphic to all its subgroups (pure subgroups, direct summands, respectively) of the same cardinality as . Obviously quasi-minimality implies pure quasi-minimality which in turn implies direct quasi-minimality, but we show that neither converse implication holds. We obtain a complete characterisation of quasi-minimal groups. In the purely quasi-minimal case, assuming GCH, a complete characterisation is also established. An independence result is proved for directly quasi-minimal groups.

     

Coloring abelian groups

A sufficient (and necessary, if n=2) condition for the existence of a particular kind of n-coloring of an abelian group is given, and applied to show that (a) the real line is colorable with two colors so that the distance 1 is forbidden for one color, and the distance s>0 for the other, or so that both 1 and s are forbidden for both colors, if and only if s is not the ratio of an odd and an even integer; (b) the chromatic number of Q² and Q³ is 2, but that of Qⁿ is greater than 2 for n>3.     

On the p-rank of Ext(A,B) for countable abelian groups A and B

In this note we show that the p-rank of Ext(A,B) for countable torsion-free abelian groups A and B is either countable or the size of the continuum.     

Sumset phenomenon in countable amenable groups

Jin proved that whenever A and B are sets of positive upper density in Z, A+B is piecewise syndetic. Jin's theorem was subsequently generalized by Jin and Keisler to a certain family of abelian groups, which in particular contains Zd. Answering a question of Jin and Keisler, we show that this result can be extended to countable amenable groups. Moreover we establish that such sumsets (or — depending on the notation — “product sets”) are piecewise Bohr, a result which for G=Z was proved by Bergelson, Furstenberg and Weiss. In the case of an abelian group G, we show that a set is piecewise Bohr if and only if it contains a sumset of two sets of positive upper Banach density.     

Sum-free sets in abelian groups

We show that there is an absolute constant δ>0 such that the number of sum-free subsets of any finite abelian groupG is