Minimal sumsets in finite solvable groups

Given a group G and positive integers r,s≤|G|, we denote by μ_G(r,s) the least possible size of a product set AB={ab∣a∈A,b∈B}, where A,B run over all subsets of G of size r,s, respectively. While the function μ_G is completely known when G is abelian [S. Eliahou, M. Kervaire, Minimal sumsets in infinite abelian groups, Journal of Algebra 287 (2005) 449-457], it is largely unknown for G non-abelian, in part because efficient tools for proving lower bounds on μ_G are still lacking in that case. Our main result here is a lower bound on μ_G for finite solvable groups, obtained by building it up from the abelian case with suitable combinatorial arguments. The result may be summarized as follows: if G is finite solvable of order m, then μ_G(r,s)≥μ_G^′(r,s), where G^′ is any abelian group of the same order m. Equivalently, with our knowledge of μ_G^′, our formula reads .One nice application is the full determination of the function μ_G for the dihedral group G=D_n and all n≥1. Up to now, only the case where n is a prime power was known. We prove that, for all n≥1, the group D_n has the same μ-function as an abelian group of order |D_n|=2n.     

Optimally small sumsets in finite abelian groups

Let G be a finite abelian group of order g. We determine, for all 1?r,s?g, the minimal size μ_G(r,s)=min|A+B| of sumsets A+B, where A and B range over all subsets of G of cardinality r and s, respectively. We do so by explicit construction. Our formula for μ_G(r,s) shows that this function only depends on the cardinality of G, not on its specific group structure. Earlier results on μ_G are recalled in the Introduction.     

Improving the Erd?s-Ginzburg-Ziv theorem for some non-abelian groups

Let G be a group of order m. Define s(G) to be the smallest value of t such that out of any t elements in G, there are m with product 1. The Erd?s-Ginzburg-Ziv theorem gives the upper bound s(G)?2m−1, and a lower bound is given by s(G)?D(G)+m−1, where D(G) is Davenport's constant. A conjecture by Zhuang and Gao [J.J. Zhuang, W.D. Gao, Erd?s-Ginzburg-Ziv theorem for dihedral groups of large prime index, European J. Combin. 26 (2005) 1053-1059] asserts that s(G)=D(G)+m−1, and Gao [W.D. Gao, A combinatorial problem on finite abelian groups, J. Number Theory 58 (1996) 100-103] has proven this for all abelian G. In this paper we verify the conjecture for a few classes of non-abelian groups: dihedral and dicyclic groups, and all non-abelian groups of order pq for p and q prime.     

Minimum product set sizes in nonabelian groups of order pq

Let G be a nonabelian group of order pq, where p and q are distinct odd primes. We analyze the minimum product set cardinality μG(r,s)=min|AB|, where A and B range over all subsets of G of cardinalities r and s, respectively. In this paper, we completely determine μG(r,s) in the case where G has order 3p and conjecture that this result can be extended to all nonabelian groups of order pq. We also prove that for every nonabelian group of order pq there exist 1?r,s?pq such that μG(r,s)>μZ/pqZ(r,s).     

On the extendability of Bi-Cayley graphs of finite abelian groups

Let G be a finite group and A a nonempty subset (possibly containing the identity element) of G. The Bi-Cayley graph X=BC(G,A) of G with respect to A is defined as the bipartite graph with vertex set G×{0,1} and edge set {{(g,0),(sg,1)}∣g∈G,s∈A}. A graph Γ admitting a perfect matching is called n-extendable if ∣V(Γ)∣≥2n+2 and every matching of size n in Γ can be extended to a perfect matching of Γ. In this paper, the extendability of Bi-Cayley graphs of finite abelian groups is explored. In particular, 2-extendable and 3-extendable Bi-Cayley graphs of finite abelian groups are characterized.     

Artin's L-functions and one-dimensional characters

Let K/Q be a finite Galois extension with the Galois group G, let χ₁,…,χr be the irreducible non-trivial characters of G, and let A be the C-algebra generated by the Artin L-functions L(s,χ₁),…,L(s,χr). Let B be the subalgebra of A generated by the L-functions corresponding to induced characters of non-trivial one-dimensional characters of subgroups of G. We prove: (1) B is of Krull dimension r and has the same quotient field as A; (2) B=A iff G is M-group; (3) the integral closure of B in A equals A iff G is quasi-M-group.     

Broué's conjecture for non-principal 3-blocks of finite groups

In representation theory of finite groups, there is a well-known and important conjecture due to M. Broué. He conjectures that, for any prime p, if a p-block A of a finite group G has an abelian defect group D, then A and its Brauer correspondent p-block B of N_G(D) are derived equivalent. We demonstrate in this paper that Broué's conjecture holds for two non-principal 3-blocks A with elementary abelian defect group D of order 9 of the O'Nan simple group and the Higman-Sims simple group. Moreover, we determine these two non-principal block algebras over a splitting field of characteristic 3 up to Morita equivalence.     

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.     

Finite 2-groups with exactly one nonmetacyclic maximal subgroup

We determine here the structure of the title groups. All such groups G will be given in terms of generators and relations, and many important subgroups of these groups will be described. Let d(G) be the minimal number of generators of G. We have here d(G) ≤ 3 and if d(G) = 3, then G′ is elementary abelian of order at most 4. Suppose d(G) = 2. Then G′ is abelian of rank ≤ 2 and G/G′ is abelian of type (2, 2^m), m ≥ 2. If G′ has no cyclic subgroup of index 2, then m = 2. If G′ is noncyclic and G/Φ(G ₀) has no normal elementary abelian subgroup of order 8, then G′ has a cyclic subgroup of index 2 and m = 2. But the most important result is that for all such groups (with d(G) = 2) we have G = AB, for suitable cyclic subgroups A and B. Conversely, if G = AB is a finite nonmetacyclic 2-group, where A and B are cyclic, then G has exactly one nonmetacyclic maximal subgroup. Hence, in this paper the nonmetacyclic 2-groups which are products of two cyclic subgroups are completely determined. This solves a long-standing problem studied from 1953 to 1956 by B. Huppert, N. Itô and A. Ohara. Note that if G = AB is a finite p-group, p > 2, where A and B are cyclic, then G is necessarily metacyclic (Huppert [4]). Hence, we have solved here problem Nr. 776 from Berkovich [1].     

Embeddings of minimal non-abelian p-groups

By a quasi-permutation matrix we mean a square matrix over the complex field C with non-negative integral trace. For a given finite group G, let p(G) denote the minimal degree of a faithful representation of G by permutation matrices, and let c(G) denote the minimal degree of a faithful representation of G by quasi-permutation matrices. See [4]. It is easy to see that c(G) is a lower bound for p(G). Behravesh [H. Behravesh, The minimal degree of a faithful quasi-permutation representation of an abelian group, Glasg. Math. J. 39 (1) (1997) 51-57] determined c(G) for every finite abelian group G and also [H. Behravesh, Quasi-permutation representations of p-groups of class 2, J. Lond. Math. Soc. (2) 55 (2) (1997) 251-260] gave the algorithm of c(G) for each finite group G. In this paper, we first improve this algorithm and then determine c(G) and p(G) for an arbitrary minimal non-abelian p-group G.     

Two classes of finite groups whose Chermak-Delgado lattice is a chain of length zero

It is an open question in the study of Chermak-Delgado lattices precisely which finite groups G have the property that 𝒞𝒟(G) is a chain of length 0. In this note, we determine two classes of groups with this property. We prove that if G = AB is a finite group, where A and B are abelian subgroups of relatively prime orders with A normal in G, then the Chermak-Delgado lattice of G equals {AC_B(A)}, a strengthening of earlier known results.     

Purity and Self-Small Groups

Let A be an abelian group. A group B is A-solvable if the natural map Hom(A, B) ?  _E(A) A → B is an isomorphism. We study pure subgroups of A-solvable groups for a self-small group A of finite torsion-free rank. Particular attention is given to the case that A is in , the class of self-small mixed groups G with G/tG? ? ⁿ for some n < ω. We obtain a new characterization of the elements of , and demonstrate that  differs in various ways from the class  ? of torsion-free abelian groups of finite rank despite the fact that the quasi-category ?  is dual to a full subcategory of ?  ?.     

Spectral Synthesis and Topologies on Ideal Spaces for Banach*-Algebras

This paper continues the study of spectral synthesis and the topologies τ_∞ and τ_r on the ideal space of a Banach algebra, concentrating on the class of Banach ^*-algebras, and in particular on L¹-group algebras. It is shown that if a group G is a finite extension of an abelian group then τ_r is Hausdorff on the ideal space of L¹(G) if and only if L¹(G) has spectral synthesis, which in turn is equivalent to G being compact. The result is applied to nilpotent groups, [FD]⁻-groups, and Moore groups. An example is given of a non-compact, non-abelian group G for which L¹(G) has spectral synthesis. It is also shown that if G is a non-discrete group then τ_r is not Hausdorff on the ideal lattice of the Fourier algebra A(G).     

On equality of central and class preserving automorphisms of finite p-groups

Let G be a finite non-abelian p-group, where p is a prime. Let Aut_c(G) and Aut_z(G) respectively denote the group of all class preserving and central automorphisms of G. We give a necessary and sufficient condition for G such that Aut_c(G) = Aut_z(G) and classify all finite non-abelian p-groups G with elementary abelian or cyclic center such that Aut_c(G) = Aut_z(G). We also characterize all finite p-groups G of order ≤ p ⁷ such that Aut_z(G) = Aut_z(G) and complete the classification of all finite p-groups of order ≤ p ⁵ for which there exist non-inner class preserving automorphisms.     

On a particular case of the inconsistent linear matrix equation AX+YB=C

We consider the linear matrix equation AX+YB=C where A,B, and C are given matrices of dimensions (r+1)×r, s×(s+1), and (r+1)×(s+1), respectively, and rank A = r, rank B = s. We give a connection between the least-squares solution and the solution which minimizes an arbitrary norm of the residual matrix C?AX? YB.     

Amenability and weak amenability of the Fourier algebra

Let G be a locally compact group. We show that its Fourier algebra A(G) is amenable if and only if G has an abelian subgroup of finite index, and that its Fourier–Stieltjes algebra B(G) is amenable if and only if G has a compact, abelian subgroup of finite index. We then show that A(G) is weakly amenable if the component of the identity of G is abelian, and we prove some partial results towards the converse.Research supported by NSERC under grant no. 90749-00.Research supported by NSERC under grant no. 227043-00.     

On maximal abelian subgroups in finite 2-groups

We determine here up to isomorphism the structure of any finite nonabelian 2-group G in which every two distinct maximal abelian subgroups have cyclic intersection. We obtain five infinite classes of such 2-groups (Theorem 1.1). This solves for p = 2 the problem Nr. 521 stated by Berkovich (in preparation). The more general problem Nr. 258 stated by Berkovich (in preparation) about the structure of finite nonabelian p-groups G such that A ∩ B = Z(G) for every two distinct maximal abelian subgroups A and B is treated in Theorems 3.1 and 3.2. In Corollary 3.3 we get a new result for an arbitrary finite 2-group. As an application of Theorems 3.1 and 3.2, we solve for p = 2 a problem of Heineken-Mann (Problem Nr. 169 stated in Berkovich, in preparation), classifying finite 2-groups G such that A/Z(G) is cyclic for each maximal abelian subgroup A (Theorem 4.1).      

Identities and isomorphisms of graded simple algebras

Let G be an arbitrary abelian group and let A and B be two finite dimensional G-graded simple algebras over an algebraically closed field F such that the orders of all finite subgroups of G are invertible in F. We prove that A and B are isomorphic if and only if they satisfy the same G-graded identities. We also describe all isomorphism classes of finite dimensional G-graded simple algebras.     

Proper partial geometries with Singer groups and pseudogeometric partial difference sets

A partial geometry admitting a Singer group G is equivalent to a partial difference set in G admitting a certain decomposition into cosets of line stabilizers. We develop methods for the classification of these objects, in particular, for the case of abelian Singer groups. As an application, we show that a proper partial geometry Π=pg(s+1,t+1,2) with an abelian Singer group G can only exist if t=2(s+2) and G is an elementary abelian 3-group of order ³(s+1) or Π is the Van Lint-Schrijver partial geometry. As part of the proof, we show that the Diophantine equation (m³−1)/2=(²rw−1)/(r²−1) has no solutions in integers m,r?1, w?2, settling a case of Goormaghtigh's equation.     

On systems of finite sets with constraints on their unions and intersections

Let F be a family of subsets of an n-element set. F is said to be of type (n, r, s) if A ∈ F, B ∈ F implies that |A ∪ B| ? n ? r, and |A ∩ B| ? s. Let f(n, r, s) = max {|F| : F is of type (n, r, s)}. We prove that f(n, r, s) ? f(n ? 1, r ? 1, s) + f(n ? 1, r + 1, s) if r > 0, n > s. And this result is used to give simple and unified proofs of Katona's and Frankl's results on f(n, r, s) when s = 0 and s = 1.