Locally Finite Groups with All Subgroups Normal-by-(Finite Rank)

A group is said to have finite (special) rank ≤ sif all of its finitely generated subgroups can be generated byselements. LetGbe a locally finite group and suppose thatH/H_Ghas finite rank for all subgroupsHofG, whereH_Gdenotes the normal core ofHinG. We prove that thenGhas an abelian normal subgroup whose quotient is of finite rank (Theorem 5). If, in addition, there is a finite numberrbounding all of the ranks ofH/H_G, thenGhas an abelian subgroup whose quotient is of finite rank bounded in terms ofronly (Theorem 4). These results are based on analogous theorems on locally finitep-groups, in which case the groupGis also abelian-by-finite (Theorems 2 and 3).     

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.     

Farrell cohomology and Brown theorems for profinite groups

LetG be a profinite group which has an open subgroupH such that the cohomologicalp-dimensiond≔cd_p(H) is finite (p is a fixed prime). The main result of this paper expresses thep-primary part of high degree cohomology ofG in terms of the elementary abelianp-subgroups ofG: From the latter one constructs a natural profinite simplicial setA G, on whichG acts by conjugation. ThenH ⁿ(G,M)≅H _G ⁿ (AG,M) holds forn≧d+r and everyp-primary discreteG-moduleM (r≔p-rank ofG). If one uses profinite Farrell cohomology, which is introduced in this paper, the analogous fact holds in all degrees. These results are the profinite analogues of theorems by K.S. Brown for discrete groups.     

Finite phase transitions in countable abelian groups

Let A be an infinite set that generates a group G. The sphere S _A (r) is the set of elements of G for which the word length with respect to A is exactly r. We say G admits all finite transitions if for every r ≥ 2 and every finite symmetric subset W ì G\{e}{W \subset G{\setminus}\{e\}}, there exists an A with S _A (r) = W. In this paper we determine which countable abelian groups admit all finite transitions. We also show that \mathbbRⁿ{\mathbb{R}^n} and the finitary symmetric group on \mathbbN{\mathbb{N}} admit all finite transitions.     

A remark on presentations of certain Chevalley groups

Let Φ be a root system of typeA _ℓ, ℓ ≧ 2,D _ℓ, ℓ ≧ 4 orE _ℓ, 6 ≧ ℓ ≧ 8 andG a group generated by nonidentity abelian subgroupsA _r,r∈Φ, satisfying:

On the number of certain subgraphs contained in graphs with a given number of edges

All graphs considered are finite, undirected, with no loops, no multiple edges and no isolated vertices. For two graphsG, H, letN(G, H) denote the number of subgraphs ofG isomorphic toH. Define also, forl≧0,N(l, H)=maxN(G, H), where the maximum is taken over all graphsG withl edges. We determineN(l, H) precisely for alll≧0 whenH is a disjoint union of two stars, and also whenH is a disjoint union ofr≧3 stars, each of sizes ors+1, wheres≧r. We also determineN(l, H) for sufficiently largel whenH is a disjoint union ofr stars, of sizess ₁≧s ₂≧…≧s _r>r, provided (s ₁−s _r)²<s ₁+s _r−2r. We further show that ifH is a graph withk edges, then the ratioN(l, H)/l ^k tends to a finite limit asl→∞. This limit is non-zero iffH is a disjoint union of stars.     

On theG-matrices with entries and eigenvalues inQ(i)

LetG be a finite abelian group,K a subfield ofC, C[G] regarded as an algebra of matrices.A _G ^K {AC[G]| all the entries and eigenvalues ofA are inK} is an association algebra overK. In this paper, the association scheme ofA _G ^K is determined and in the caseK=Q(i), the first eigenmatrix of the association scheme computed. As an application, it is proved thatZ ₄×Z ₄×Z ₄ is the only abelian group admitted as a Singer group by some distance-regular digraph of girth 4 on 64 vertices.     

Ramsey families which exclude a graph

For graphsA andB the relationA(B) _r ¹ means that for everyr-coloring of the vertices ofA there is a monochromatic copy ofB inA. Forb (G) is the family of graphs which do not embedG. A familyof graphs is Ramsey if for all graphsBthere is a graphAsuch thatA(B) _r ¹ . The only graphsG for which it is not known whether Forb (G) is Ramsey are graphs which have a cutpoint adjacent to every other vertex except one. In this paper we prove for a large subclass of those graphsG, that Forb (G) does not have the Ramsey property.This research has been supported in part by NSERC grant 69-1325.     

A measuring argument for finite permutation groups

LetG be a finite transitive permutation group on a finite setS. LetA be a nonempty subset ofS and denote the pointwise stabilizer ofA inG byC _G (A). Our main result is the following inequality: [G :C _G (A)]≥|G|^|A|/|S|. This paper is a part of the author’s Ph.D. thesis research, carried out at Tel Aviv University under the supervision of Professor Marcel Herzog.     

The algebra ofG-relations

In this paper, we study a tower {A _n ^G: n} ≥ 1 of finite-dimensional algebras; here, G represents an arbitrary finite group,d denotes a complex parameter, and the algebraA _n ^G(d) has a basis indexed by ‘G-stable equivalence relations’ on a set whereG acts freely and has 2n orbits. We show that the algebraA _n ^G(d) is semi-simple for all but a finite set of values ofd, and determine the representation theory (or, equivalently, the decomposition into simple summands) of this algebra in the ‘generic case’. Finally we determine the Bratteli diagram of the tower {A _n ^G(d): n} ≥ 1 (in the generic case).     

Stability for large forbidden subgraphs

In this note we strengthen the stability theorem of Erd?s and Simonovits. Write K_r(s₁, …, s_r) for the complete r‐partite graph with classes of sizes s₁, …, s_r and T_r(n) for the r‐partite Turán graph of order n. Our main result is: For all r≥2 and all sufficiently small c>0, ε>0, every graph G of sufficiently large order n with e(G)>(1?1/r?ε)n²/2 satisfies one of the conditions:

• (a) G contains a $K_{r+1} (\lfloor c\,\mbox{ln}\,n \rfloor,\ldots,\lfloor c\,\mbox{ln}\,n \rfloor,\lceil n^{{1}-\sqrt{c}}\rceil)In this note we strengthen the stability theorem of Erd?s and Simonovits. Write K_r(s₁, …, s_r) for the complete r‐partite graph with classes of sizes s₁, …, s_r and T_r(n) for the r‐partite Turán graph of order n. Our main result is: For all r≥2 and all sufficiently small c>0, ε>0, every graph G of sufficiently large order n with e(G)>(1?1/r?ε)n²/2 satisfies one of the conditions:
  • (a) G contains a $K_{r+1} (\lfloor c\,\mbox{ln}\,n \rfloor,\ldots,\lfloor c\,\mbox{ln}\,n \rfloor,\lceil n^{{1}-\sqrt{c}}\rceil)$;
  • (b) G differs from T_r(n) in fewer than (ε^1/3+c^1/(3r+3))n² edges.
  Letting µ(G) be the spectral radius of G, we prove also a spectral stability theorem: For all r≥2 and all sufficiently small c>0, ε>0, every graph G of sufficiently large order n with µ(G)>(1?1/r?ε)n satisfies one of the conditions:
  • (a) G contains a $K_{r+1}(\lfloor c\,\mbox{ln}\,n\rfloor,\ldots,\lfloor c\,\mbox{ln}\,n\rfloor,\lceil n^{1-\sqrt{c}}\rceil)$;
  • (b) G differs from T_r(n) in fewer than (ε^1/4+c^1/(8r+8))n² edges.
  © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 362–368, 2009     

Additive Latin transversals and group rings

LetA={a ₁, …,a _k} and {b ₁, …,b _k} be two subsets of an abelian groupG, k≤|G|. Snevily conjectured that, when |G| is odd, there is a numbering of the elements ofB such thata _i+b _i,1≤i≤k are pairwise distinct. By using a polynomial method, Alon affirmed this conjecture for |G| prime, even whenA is a sequence ofk<|G| elements. With a new application of the polynomial method, Dasgupta, Károlyi, Serra and Szegedy extended Alon’s result to the groupsZ _p ^r andZ _p ^rin the casek<p and verified Snevily’s conjecture for every cyclic group. In this paper, by employing group rings as a tool, we prove that Alon’s result is true for any finite abelianp-group withk<√2p, and verify Snevily’s conjecture for every abelian group of odd order in the casek<√p, wherep is the smallest prime divisor of |G|. This work has been supported partly by NSFC grant number 19971058 and 10271080.     

Weakly symmetric spaces and bounded symmetric domains

LetG be a connected, simply-connected, real semisimple Lie group andK a maximal compactly embedded subgroup ofG such thatD=G/K is a hermitian symmetric space. Consider the principal fiber bundleM=G/K _s G/K, whereK _s is the semisimple part ofK=K _s ·Z _K ⁰ andZ _K ⁰ is the connected center ofK. The natural action ofG onM extends to an action ofG ¹=G×Z _K ⁰ . We prove as the main result thatM is weakly symmetric with respect toG ¹ and complex conjugation. In the case whereD is an irreducible classical bounded symmetric domain andG is a classical matrix Lie group under a suitable quotient, we provide an explicit construction ofM=D×S ¹ and determine a one-parameter family of Riemannian metrics onM invariant underG ¹. Furthermore,M is irreducible with respect to . As a result, this provides new examples of weakly symmetric spaces that are nonsymmetric, including those already discovered by Selberg (cf. [M]) for the symplectic case and Berndt and Vanhecke [BV1] for the rank-one case.Research partially supported by an NSF grant. The author wishes to thank the International Erwin Schroedinger Institute for its hospitality during the preparation of this paper.     

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.     

Fractals related to Pascal's triangle

A precise definition of a fractalF _p ^{r 1} derived from Pascal's triangle modulop ^r (p prime) is given. The number of nonzero terms in the firstp ^s lines of Pascal's triangle modulop ^r is computed. From this result the Hausdorff dimension and Hausdorff measure ofF _p ^{r 1} are deduced. The nonself-similarty ofF _p ^{r 1},r2, is also discussed.     

Symmetric colorings of the dihedral group

Let G be a finite group and let r∈?. An r-coloring of G is any mapping χ:G→{1,…,r}. Colorings χ and ψ are equivalent if there exists g∈G such that χ(xg^?1) = ψ(x) for every x∈G. A coloring χ is symmetric if there exists g∈G such that χ(gx^?1g) = χ(x) for every x∈G. Let S_r(G) denote the number of symmetric r-colorings of G and s_r(G) the number of equivalence classes of symmetric r-colorings of G. We count S_r(G) and s_r(G) in the case where G is the dihedral group D_n.     

Graded rings and essential ideals

LetG be a group andA aG-graded ring. A (graded) idealI ofA is (graded) essential ifI⊃J≠0 wheneverJ is a nonzero (graded) ideal ofA. In this paper we study the relationship between graded essential ideals ofA, essential ideals of the identity componentA _e and essential ideals of the smash productA#G ^*. We apply our results to prime essential rings, irredundant subdirect sums and essentially nilpotent rings.     

On the set of discrete subgroups of bounded covolume in a semisimple group

In this noteG is a locally compact group which is the product of finitely many groups G_s(k_s)(s∈S), where k_s is a local field of characteristic zero and G_s an absolutely almost simplek _s-group, ofk _s-rank ≥1. We assume that the sum of the r_s is ≥2 and fix a Haar measure onG. Then, given a constantc > 0, it is shown that, up to conjugacy,G contains only finitely many irreducible discrete subgroupsL of covolume ≥c (4.2). This generalizes a theorem of H C Wang for real groups. His argument extends to the present case, once it is shown thatL is finitely presented (2.4) and locally rigid (3.2).     

On finite groups whose every proper normal subgroup is a union of a given number of conjugacy classes

Let G be a finite group andA be a normal subgroup ofG. We denote by ncc(A) the number ofG-conjugacy classes ofA andA is calledn-decomposable, if ncc(A)= n. SetK _G = {ncc(A)|A ⊲ G}. LetX be a non-empty subset of positive integers. A groupG is calledX-decomposable, ifK _G =X. Ashrafi and his co-authors [1-5] have characterized theX-decomposable non-perfect finite groups forX = {1, n} andn ≤ 10. In this paper, we continue this problem and investigate the structure ofX-decomposable non-perfect finite groups, forX = {1, 2, 3}. We prove that such a group is isomorphic to Z₆, D₈, Q₈, S₄, SmallGroup(20, 3), SmallGroup(24, 3), where SmallGroup(m, n) denotes the mth group of ordern in the small group library of GAP [11].     

The character table of the group GL 2(Q) when extended by a certain group of order two

LetG denote either of the groupsGL ₂(q) or SL₂(q). Then θ :G →G given by θ(A) = (A ^t)^t, whereA ^t denotes the transpose of the matrixA, is an automorphism ofG. Therefore we may form the groupG.θ> which is the split extension of the groupG by the cyclic group θ of order 2. Our aim in this paper is to find the complex irreducible character table ofG. θ.