Discreteness criterion for subgroups of products of SL(2)

Let G be a finite product of SL(2, K _i )’s for local fields K _i of characteristic zero. We present a discreteness criterion for nonsolvable subgroups of G containing an irreducible lattice of a maximal unipotent subgroup of G. In particular, such a subgroup has to be arithmetic. This extends a previous result of A. Selberg when G is a product of SL₂( \mathbbR \mathbb{R} )’s.     

Blocks of full defect with nonabelian metacyclic defect groups

Let p be an odd prime and let B be a p-block of a finite group G with a nonabelian metacyclic defect group P which is a Sylow p-subgroup of G. The purpose of this article is to study the ordinary and modular irreducible characters in B. In particular, we calculate k _i (B) and l _i (B) for an arbitrary nonnegative integer i.     

Weak Cayley Table Groups II: Alternating Groups and Finite Coxeter Groups

A weak Cayley table isomorphism is a bijection φ: G → H of groups such that φ(xy) ～ φ(x)φ(y) for all x, y ∈ G. Here ～denotes conjugacy. When G = H the set of all weak Cayley table isomorphisms φ: G → G forms a group 𝒲(G) that contains the automorphism group Aut(G) and the inverse map I: G → G, x → x ^?1. Let 𝒲₀(G) = ?Aut(G), I? ≤ 𝒲(G) and say that G has trivial weak Cayley table group if 𝒲(G) = 𝒲₀(G). We show that all finite irreducible Coxeter groups (except possibly E ₈) have trivial weak Cayley table group, as well as most alternating groups. We also consider some sporadic simple groups.     

Sums of lengtht in Abelian groups

LetG be an Abelian group written additively,B a finite subset ofG, and lett be a positive integer. Fort≦|B|, letB _t denote the set of sums oft distinct elements overB. Furthermore, letK be a subgroup ofG and let σ denote the canonical homomorphism σ:G→G/K. WriteB _t (modB _t) forB _tσ and writeB _t (modK) forBσ. The following addition theorem in groups is proved. LetG be an Abelian group with no 2-torsion and letB a be finite subset ofG. Ift is a positive integer such thatt<|B| then |B _t (modK)|≧|B (modK)| for any finite subgroupK ofG.     

Groups containing a strongly embedded subgroup

An involution v of a group G is said to be finite (in G) if vv ^g has finite order for any g ∈ G. A subgroup B of G is called a strongly embedded (in G) subgroup if B and G\B contain involutions, but B ∩ B ^g does not, for any g ∈ G\B. We prove the following results. Let a group G contain a finite involution and an involution whose centralizer in G is periodic. If every finite subgroup of G of even order is contained in a simple subgroup isomorphic, for some m, to L ₂(2 ^m ) or Sz(2 ^m ), then G is isomorphic to L ₂(Q) or Sz(Q) for some locally finite field Q of characteristic two. In particular, G is locally finite (Thm. 1). Let a group G contain a finite involution and a strongly embedded subgroup. If the centralizer of some involution in G is a 2-group, and every finite subgroup of even order in G is contained in a finite non-Abelian simple subgroup of G, then G is isomorphic to L ₂(Q) or Sz(Q) for some locally finite field Q of characteristic two (Thm. 2). Supported by RFBR (project No. 08-01-00322), by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-334.2008.1), and by the Russian Ministry of Education through the Analytical Departmental Target Program (ADTP) “Development of Scientific Potential of the Higher School of Learning” (project Nos. 2.1.1.419 and 2.1.1./3023). (D. V. Lytkina and V. D. Mazurov) Translated from Algebra i Logika, Vol. 48, No. 2, pp. 190–202, March–April, 2009.     

On r-recognition by prime graph of Bp(3) where p is an odd prime

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p and p′ are joined by an edge if there is an element in G of order pp′. We denote by k(Γ(G)) the number of isomorphism classes of finite groups H satisfying Γ(G) = Γ(H). Given a natural number r, a finite group G is called r-recognizable by prime graph if k(Γ(G)) =  r. In Shen et al. (Sib. Math. J. 51(2):244–254, 2010), it is proved that if p is an odd prime, then B _p (3) is recognizable by element orders. In this paper as the main result, we show that if G is a finite group such that Γ(G) = Γ(B _p (3)), where p > 3 is an odd prime, then G @ B_p(3){G\cong B_p(3)} or C _p (3). Also if Γ(G) = Γ(B ₃(3)), then G @ B₃(3), C₃(3), D₄(3){G\cong B_3(3), C_3(3), D_4(3)}, or G/O₂(G) @ Aut(²B₂(8)){G/O_2(G)\cong {\rm Aut}(^2B_2(8))}. As a corollary, the main result of the above paper is obtained.     

On locally finite groups with a small centralizer of a four-subgroup

The main result of the paper is the following theorem. Let G be a locally finite group having a four-subgroup V such that C _G (V) is finite. Suppose that V contains two involutions v ₁ and v ₂ such that the centralizers C _G (v ₁) and C _G (v ₂) have finite exponent. Then G is almost locally soluble and [G, V]′ has finite exponent. Since [G, V] has finite index in G, the result gives a fairly detailed information about the structure of G.     

Automorphismen nilpotenter proendlicher Gruppen

The object of this paper is to construct outer automorphisms for nilpotent torsion-free profinite groups, what reduces to nilpotent compact torsion-free pro-p-groups. If the class of such a group is greater than 2, a characteristic subgroupA can be found, such that the stabilizer of the series 1AC _G (A)G in AutG contains a subgroup which modulo InnG is isomorphic to the center ofG. A similar result is obtained for class 2 groups with some exception for which AutG is given explicitely. In course of the proof for the class 2 case, the group Hom (A, B) of continuous homomorphisms is analysed, whereA is a locally direct product of a family {A _i ;iI} and allA _i 's andB are locally compact abelian groups.     

Ki-covers. II. Ki-perfect graphs

AK_i is a complete subgraph of size i. A K_i-cover of a graph G(V, E) is a set C of K_i?1s of G such that every K_i in G contains at least one K_i?1 in C . c_i(G) is the cardinality of a smallest K_i-cover of G. A K_i-packing of G is a set of K_is such that no two K_is have i ? 1 nodes in common. p_i(G) is the cardinality of a largest K_i-packing of G. Let F _i(G) denote the set of K_is in G and define c_i(F) and p_i(F) analogously for F ? F _i(G). G is K_i-perfect if ?F ? F _i(G), c_i(F) = p_i(F). The K₂-perfect graphs are precisely the bipartite graphs. We present a characterization of K_i-perfect graphs that is similar to the Strong Perfect Graph Conjecture, and explore the relationships between K_i-perfect graphs and normal hypergraphs. Furthermore, if iA denotes the 0 ? 1 matrix of G where the rows are the elements of F _i?1(G) that belong to at least one K_i and the columns are the elements of F _i(G), then we show that iA is perfect iff G is a K_i-perfect graph. We also characterize the K_i-perfect graphs for which iAis balanced.     

Iterated clique graphs with increasing diameters

A simple argument by Hedman shows that the diameter of a clique graph G differs by at most one from that of K(G), its clique graph. Hedman described examples of a graph G such that diam(K(G)) = diam(G) + 1 and asked in general about the existence of graphs such that diam(Kⁱ(G)) = diam(G) + i. Examples satisfying this equality for i = 2 have been described by Peyrat, Rall, and Slater and independently by Balakrishnan and Paulraja. The authors of the former work also solved the case i = 3 and i = 4 and conjectured that such graphs exist for every positive integer i. The cases i ≥ 5 remained open. In the present article, we prove their conjecture. For each positive integer i, we describe a family of graphs G such that diam(Kⁱ(G)) = diam(G) + i. © 1998 John Wiley & Sons, Inc. J. Graph Theory 28: 147–154, 1998     

On normal subgroups in the periodic products of S.I. Adian

A subgroup H of a given group G is called a hereditarily factorizable subgroup (HF subgroup) if each congruence on H can be extended to some congruence on the entire group G. An arbitrary group G ₁ is an HF subgroup of the direct product G ₁ × G ₂, as well as of the free product G ₁ * G ₂ of groups G ₁ and G ₂. In this paper a necessary and sufficient condition is obtained for a factor Gi of Adian’s n-periodic product Π _i∈I ⁿ G _i of an arbitrary family of groups {G _i } _i∈I to be an HF subgroup. We also prove that for each odd n ≥ 1003 any noncyclic subgroup of the free Burnside group B(m, n) contains an HF subgroup isomorphic to the group B(∞, n) of infinite rank. This strengthens the recent results of A.Yu. Ol’shanskii and M. Sapir, D. Sonkin, and S. Ivanov on HF subgroups of free Burnside groups. This result implies, in particular, that each (noncyclic) subgroup of the group B(m, n) is SQ-universal in the class of all groups of period n. Moreover, it turns out that any countable group of period n is embedded in some 2-generated group of period n, which strengthens the previously obtained result of V. Obraztsov. At the end of the paper we prove that the group B(m, n) is distinguished as a direct factor in any n-periodic group in which it is contained as a normal subgroup.     

Expanders,rank and graphs of groups

LetG be a finitely presented group, and let {G _i } be a collection of finite index normal subgroups that is closed under intersections. Then, we prove that at least one of the following must hold: 1.G _i is an amalgamated free product or HNN extension, for infinitely manyi; 2. the Cayley graphs ofG/G _i (with respect to a fixed finite set of generators forG) form an expanding family; 3. inf_i(d(G _i )−1)/[G:G _i ]=0, whered(G _i ) is the rank ofG _i . The proof involves an analysis of the geometry and topology of finite Cayley graphs. Several applications of this result are given.     

The Derived Picard Group is a Locally Algebraic Group

Let A be a finite-dimensional algebra over an algebraically closed field K. The derived Picard group DPic _K (A) is the group of two-sided tilting complexes over A modulo isomorphism. We prove that DPic _K (A) is a locally algebraic group, and its identity component is Out⁰ _K (A). If B is a derived Morita equivalent algebra then DPic _K (A)DPic _K (B) as locally algebraic groups. Our results extend, and are based on, work of Huisgen-Zimmermann, Saorín and Rouquier.     

Bogomolov multipliers for some <Emphasis Type="Italic">p</Emphasis>-groups of nilpotency class 2

The Bogomolov multiplier B ₀(G) of a finite group G is defined as the subgroup of the Schur multiplier consisting of the cohomology classes vanishing after restriction to all abelian subgroups of G. The triviality of the Bogomolov multiplier is an obstruction to Noether’s problem. We show that if G is a central product of G ₁ and G ₂, regarding K _i ≤ Z(G _i ), i = 1, 2, and θ: G ₁ → G ₂ is a group homomorphism such that its restriction \(\theta {|_{{K_1}}}:{K_1} \to {K_2}\) is an isomorphism, then the triviality of B ₀(G ₁/K ₁),B ₀(G ₁) and B ₀(G ₂) implies the triviality of B ₀(G). We give a positive answer to Noether’s problem for all 2-generator p-groups of nilpotency class 2, and for one series of 4-generator p-groups of nilpotency class 2 (with the usual requirement for the roots of unity).     

Extensions and the Induced Characters of Cyclic p-Groups

For a prime p, we denote by B_n the cyclic group of order pⁿ. Let φ be a faithful irreducible character of B_n, where p is an odd prime. We study the p-group G containing B_n such that the induced character φ^G is also irreducible. The purpose of this article is to determine the subgroup N_G(N_G(B_n)) of G under the hypothesis [N_G(B_n):B_n]⁴ ≦ pⁿ.     

On B(n,k) 2-groups

A finite group G is said to be a B(n, k) group if for any n-element subset {a ₁,…, a _n } of G, |{a _i a _j |1 ≤ i, j ≤ n}| ≤k. In this article, we give characterizations of the B(5, 19) 2-groups, and the B(6, k) 2-groups for 21 ≤ k ≤ 28.     

Infinite dimensional linear groups with many G — invariant subspaces

Let F be a field, A be a vector space over F, GL(F, A) be the group of all automorphisms of the vector space A. A subspace B of A is called nearly G-invariant, if dim _F (BFG/B) is finite. A subspace B is called almost G-invariant, if dim _F (B/Core _G (B)) is finite. In the current article, we study linear groups G such that every subspace of A is either nearly G-invariant or almost G-invariant in the case when G is a soluble p-group where p = char F.     

The Multiplicity Free Permutation Representations of the Ree Groups 2 G 2(q), the Suzuki Groups 2 B 2(q), and Their Automorphism Groups

Abstract

Let G a simple group of type ² B ₂(q) or ² G ₂(q), where q is an odd power of 2 or 3, respectively. The main goal of this paper is to determine the multiplicity free permutation representations of G and A ≤ Aut(G) where A is a subgroup containing a copy of G. Let B be a Borel subgroup of G. If G = ² B ₂(q) we show that there is only one non-trivial multiplicity free permutation representation, namely the representation of G associated to the action on G/B. If G = ² G ₂(q) we show that there are exactly two such non-trivial representations, namely the representations of G associated to the action on G/B and the action on G/M, where M = UC with U the maximal unipotent subgroup of B and C the unique subgroup of index 2 in the maximal split torus of B. The multiplicity free permutation representations of A correspond to the actions on A/H where H is isomorphic to a subgroup containing B if G = ² B ₂(q), and containing M if G = ² G ₂(q). The problem of determining the multiplicity free representations of the finite simple groups is important, for example, in the classification of distance-transitive graphs.     

Some Results on p-Nilpotence and Supersolvability of Finite Groups

Let G be a finite group. A subgroup K of a group G is called an ?-subgroup of G if N _G (K) ∩ K ^x  ≦ K for all x ? G. The set of all ?-subgroups of G will be denoted by ?(G). Let P be a nontrivial p-group. A chain of subgroups 1 = P ₀ ? P ₁ ? ··· ? P _n  = P is called a maximal chain of P provided that |P _i : P _i?1| = p, i = 1, 2, ···, n. A nontrivial p-subgroup P of G is called weakly supersolvably embedded in G if P has a maximal chain 1 = P ₀ ? P ₁ ? ··· ? P _i  ? ··· ? P _n  = P such that P _i  ? ?(G) for i = 1, 2, ···, n. Using the concept of weakly supersolvably embedded, we obtain new characterizations of p-nilpotent and supersolvable finite groups.