Minimal sumsets in finite solvable groups

Addition sets in a finite group

Let G be a group and let x, y ∈ G. If every element of G can be written as a word $${x^{{\alpha _1}}}{y^{{\alpha _2}}}{x^{{\alpha _3}}} \ldots {x^{{\alpha _{n - 1}}}}{y^{{\alpha _n}}}$$ (1) where αi ∈ ℤ, 1 ≤ i ≤ n, then we say that x and y generate G and that G is a 2-generator group. Although cyclic groups are 2-generator groups according to this definition we are only interested here in 2-generator groups which cannot be generated by a single element. Even among finite groups G many are not 2-generator groups; for example the abelian group of order 8 in which every element has order 2 cannot be 2-generated since given any pair of distinct non-trivial elements x, y there are only 4 words given by expressions of the form (1). However, many groups are 2-generated, in particular finite simple groups.

Maximal sum-free sets in finite Abelian groups, III

Let G be an Abelian group of order n, let k 2 be an integer, and A1, . . ., A k be non-empty subsets of G.and maximal by inclusion if for any i {1, . . ., k} and x G \ Ai, the collection (A1, . . ., Ai-1, Ai {x}, Ai+1, . .In this work, we study the problem of the maximal value of k (G).In particular, the maximal value of k (Z d ) for the cyclic group Z d is determined.Upper and lower bounds for k (G) are obtained for the Abelian group G.The structure of maximal k-sum-free set by capacity (by inclusion) is described for an arbitrary cyclic group.

Let $\Lg = \{+,-,0\}$ be the language of the abelian groups, $L$ an expansion of $\Lg(<)$ by relations and constants, and $\Lmod = \Lg \cup \{\equiv_n\}_{n \geq 2}$ where each $\equiv_n$ is defined as follows: $x \equiv_n y$ if and only if $n|x-y$. Let $H$ be a structure for $L$ such that $H|\Lg(<)$ is a totally ordered abelian group and $K$ a totally ordered abelian group. We consider a product interpretation of $H \times K$ with a new predicate $I$ for $\{0\}\times K$ defined by N.~Suzuki \cite{Sz}. Suppose that $H$ admits quantifier elimination in $L$. 1. If $K$ is a Presburger arithmetic with smallest positive element $1_K$ then the product interpretation $G$ of $H \times K$ with a new predicate $I$ admits quantifier elimination in $L(I, 1) \cup \Lmod$ with $1^G = (0^H, 1_K)$.2. If $K$ is dense regular and $K/nK$ is finite for every integer $n \geq 2$ then the product interpretation $G$ of $H \times K$ with a new predicate $I$ admits quantifier elimination in $L(I, D) \cup \Lmod$ for some set $D$ of constant symbols where $G \models I(d)$ for each $d \in D$.3. If $K$ admits quantifier elimination in $\Lmod(<, D)$ for some set $D$ of constant symbols then the product interpretation $G$ of $H \times K$ with a new predicate $I$ admits quantifier elimination in $L(I, D) \cup \Lmod$ unless $K$ is dense regular with $K/nK$ being infinite for some $n$. Conversely, if the product interpretation $G$ of $H \times K$ with a new predicate $I$ admits quantifier elimination in $L(I, D) \cup \Lmod$ for some set $D$ of constant symbols such that $G \models I(d)$ for each $d \in D$ then $H$ admits quantifier elimination in $L \cup \Lmod$, and $K$ admits quantifier elimination in $\Lmod(<, D)$. We also discuss the axiomatization of the theory of the product interpretation of $H \times K$. %For some set $C$ of constants in $K$.

A compactness argument in the additive theory and the polynomial method

The homological functors of a group were first introduced in homotopy theory. Some of the homological functors including the nonabelian tensor square and the Schur multiplier of abelian groups of prime power order are determined in this paper. The nonabelian tensor square of a group G introduced by Brown and Loday in 1987 is a special case of the nonabelian tensor product. Meanwhile, the Schur multiplier of G is the second cohomology with integer coefficients is named after Issai Schur. The aims of this paper are to determine the nonabelian tensor square and the Schur multiplier of abelian groups of order p5, where p is an odd prime

Many of the sporadic simple groups possess highly symmetric generating sets which can often be used to construct the groups, and which carry much information about their subgroup structure. We give a survey of results obtained so far.

Let Q be a quasigroup. For $$\alpha ,\beta \in S_Q$$ let $$Q_{\alpha ,\beta }$$ be the principal isotope $$x*y = \alpha (x)\beta (y)$$ . Put $$\mathbf a(Q)= |\{(x,y,z)\in Q^3;$$ $$x(yz)) = (xy)z\}|$$ and assume that $$|Q|=n$$ . Then $$\sum _{\alpha ,\beta }\mathbf a(Q_{\alpha ,\beta })/(n!)^2 = n^2(1+(n-1)^{-1})$$ , and for every $$\alpha \in S_Q$$ there is $$\sum _\beta \mathbf a(Q_{\alpha ,\beta })/n! = n(n-1)^{-1}\sum _x(f_x^2-2f_x+n)\ge n^2$$ , where $$f_x=|\{y\in Q;$$ $$ y = \alpha (y)x\}|$$ . If G is a group and $$\alpha $$ is an orthomorphism, then $$\mathbf a(G_{\alpha ,\beta })=n^2$$ for every $$\beta \in S_Q$$ . A detailed case study of $$\mathbf a(G_{\alpha ,\beta })$$ is made for the situation when $$G = \mathbb Z_{2d}$$ , and both $$\alpha $$ and $$\beta $$ are “natural” near-orthomorphisms. Asymptotically, $$\mathbf a(G_{\alpha ,\beta })>3n$$ if G is an abelian group of order n. Computational results: $$\mathbf a(7) = 17$$ and $$\mathbf a(8) \le 21$$ , where $$\mathbf a(n) = \min \{\mathbf a(Q);$$ $$ |Q|=n\}$$ . There are also determined minimum values for $$\mathbf a(G_{\alpha ,\beta })$$ , G a group of order $$\le 8$$ .

Recently, it was proved that every p-Schur ring over an abelian group of order <TEX>$p^3$</TEX> is Schurian. In this paper, we prove that every commutative p-Schur ring over a non-abelian group of order <TEX>$p^3$</TEX> is Schurian.

New constructions of divisible designs

In this paper we classify, up to equivalence, all semisimple nontrivial Hopf algebras of dimension 22n+1 for n ≥ 2 over an algebraically closed field of characteristic 0 with the group of group-like elements isomorphic to \(\mathbb {Z}_{2^{n}}\times \mathbb {Z}_{2^{n}}\). Moreover we classify all such nonisomorphic Hopf algebras of dimension 32 and show that they are not twist-equivalent to each other. More generally, given an abelian group of order 2 m−1 we give an upper bound for the number of nonisomorphic nontrivial Hopf algebras of dimension 2 m which have this group as their group of group-like elements.

Let G = (V, E) be a simple graph with n vertices, and let A be an abelian group of order n. Let f: V(G) → A be a bijection, and define the weight of a vertex x by w(x) = Σy ∈ N(x) f(y), where N(x) denotes the open neighborhood of x. We call f an A-distance magic labeling if all vertex weights are equal, and an A-distance antimagic labeling if the weights of distinct vertices are pairwise different. The prime graph of a commutative ring R is the simple graph with vertex set R in which two vertices x and y are adjacent if and only if xRy = {0}. In this paper, we investigate the existence of A-distance magic and A-distance antimagic labelings of the prime graph over the ring Zn for several values of n.

A generalization of Kraemer's result on difference sets

The aim of this paper is to prove certain characterization theorems for groups in which permutability is a transitive relation, the so called $${\mathcal{PT}}$$ -groups. In particular, it is shown that the finite solvable $${\mathcal{PT}}$$ -groups, the finite solvable groups in which every subnormal subgroup of defect two is permutable, the finite solvable groups in which every normal subgroup is permutable sensitive, and the finite solvable groups in which conjugate-permutability and permutability coincide are all one and the same class. This follows from our main result which says that the finite modular p-groups, p a prime, are those p-groups in which every subnormal subgroup of defect two is permutable or, equivalently, in which every normal subgroup is permutable sensitive. However, there exist finite insolvable groups which are not $${\mathcal{PT}}$$ -groups but all subnormal subgroups of defect two are permutable.