Generalized frobenius groups. II

A pair (G, K) in whichG is a finite group andK a normal nontrivial proper subgroup ofG is said to be an F2-pair (a Frobenius type pair) if |C _G (x)|=|C _G/K (xK)| for allx∈G\K. A theorem of Camina asserts that in this case eitherK orG/K is ap-group or elseG is a Frobenius group with Frobenius kernelK. The structure ofG will be described here under certain assumptions on the Sylowp-subgroups ofG. This author’s research was partially supported by the Technion V.P.R. fund — E.L.J. Bishop research fund. This author’s research was partially supported by the MPI fund.     

Large centralizers in finite solvable groups

In 1955 R. Brauer and K. A. Fowler showed that ifG is a group of even order >2, and the order |Z(G)| of the center ofG is odd, then there exists a strongly real) elementx∈G−Z whose centralizer satisfies|C _G(x)|>|G|^1/3. In Theorem 1 we show that every non-abeliansolvable groupG contains an elementx∈G−Z such that|C _G(x)|>[G:G′∩Z]^1/2 (and thus|C _G(x)|>|G|^1/3). We also note that if non-abelianG is either metabelian, nilpotent or (more generally) supersolvable, or anA-group, or any Frobenius group, then|C _G(x)|>|G|^1/2 for somex∈G−Z. In Theorem 2 we prove that every non-abelian groupG of orderp ^mqⁿ (p, q primes) contains a proper centralizer of order >|G|^1/2. Finally, in Theorem 3 we show that theaverage |C(x)|, x∈G, is ≧c|G| ^1/3 for metabelian groups, wherec is constant and the exponent 1/3 is best possible.     

Torsion in Boundary Coinvariants and K -Theory for Affine Buildings

Let (G, I, N, S) be an affine topological Tits system, and let Γ be a torsion-free cocompact lattice in G. This article studies the coinvariants H ₀(Γ; C(Ω,Z)), where Ω is the Furstenberg boundary of G. It is shown that the class [1] of the identity function in H ₀(Γ; C(Ω, Z)) has finite order, with explicit bounds for the order. A similar statement applies to the K ₀ group of the boundary crossed product C ^*-algebra C(Ω)Γ. If the Tits system has type ? ₂, exact computations are given, both for the crossed product algebra and for the reduced group C ^*-algebra.     

On the Edge-forwarding Indices of Frobenius Graphs

A G-Frobenius graph F, as defined by Fang, Li, and Praeger, is a connected orbital graph of a Frobenius group G = K × H with Frobenius kernel K and Frobenius complement H. F is also shown to be a Cayley graph, F = Cay（K, S） for K and some subset S of the group K. On the other hand, a network N with a routing function R, written as （N, R）, is an undirected graph N together with a routing R which consists of a collection of simple paths connecting every pair of vertices in the graph. The edge-forwarding index π（N） of a network （N, R）, defined by Heydemann, Meyer, and Sotteau, is a parameter to describe tile maximum load of edges of N. In this paper, we study the edge-forwarding indices of Frobenius graphs. In particular, we obtain the edge-forwarding index of a G-Frobenius graph F with rank（G） ≤ 50.     

Lower bounds for the number of conjugacy classes in finite solvable groups

We prove first that if G is a finite solvable group of derived length d ≥ 2, then k(G) > |G|^1/(2^d−1), where k(G) is the number of conjugacy classes in G. Next, a growth assumption on the sequence [G⁽ⁱ⁾: G⁽ⁱ⁺¹⁾] ₁ ^d−1 , where G⁽ⁱ⁾ is theith derived group, leads to a |G|^1/(2d−1) lower bound for k(G), from which we derive a |G|^c/log ₂log₂|G| lower bound, independent of d(G). Finally, “almost logarithmic” lower bounds are found for solvable groups with a nilpotent maximal subgroup, and for all Frobenius groups, solvable or not.     

Groups with an almost regular involution

An involution j of a group G is said to be almost perfect in G if any two involutions in j^G whose product has infinite order are conjugated by a suitable involution in j^G. Let G contain an almost perfect involution j and |C_G(j)| < ∞. Then the following statements hold: (1) [j,G] is contained in an FC-radical of G, and |G: [j,G]| ⩽ |C_G(j)|; (2) the commutant of an FC-radical of G is finite; (3) FC(G) contains a normal nilpotent class 2 subgroup of finite index in G. __________ Translated from Algebra i Logika, Vol. 46, No. 3, pp. 360–368, May–June, 2007.     

Some infinite groups with strongly embedded subgroup

An involution i of a group G is said to be finite if |ii^g|<∞ for all g ∃ G. Suppose that G contains a finite involution and an infinite elementary Abelian 2-subgroup S and, moreover, the normalizer H=N_G(S)=SλT is strongly embedded in G and is a Frobenius group with locally cyclic complement T. It is proved that G is isomorphic to L₂(Q) over a locally finite field Q of characteristic 2. In particular, part (a) of Question 10.76 raised by Shunkkov in the Kourovka Notebook is answered in the affirmative. Supported by RFFR grant No. 99-01-00542. Translated fromAlgebra i Logika, Vol. 39, No. 5, pp. 602–617, September–October, 2000.     

Existence ofN-INJECTORS in a not central normal Fitting class

If we denote byL(G) the semisimple radical of a groupG, we prove in this paper that ℒ=G|G=C_G(L(G))L(G) is a not central normal Fitting class. Moreover, all ( haveN-groups haveN-injectors.     

On aschbacher’s localC(G; T) theorem

Aschbacher’s localC(G; T) theorem asserts that ifG is a finite group withF*(G)=O ₂(G), andTεSyl₂(G), thenG=C(G; T)K(G), whereC(G; T)=〈N _G (T ₀)|1≠T ₀ charT〉 andK(G) is the product of all near components ofG of typeL ₂(2 ⁿ ) orA ₂ ⁿ ₊₁. Near components are also known asχ-blocks or Aschbacher blocks. In this paper we give a proof of Aschbacher’s theorem in the case thatG is aK-group, i.e., in the case that every simple section ofG is isomorphic to one of the known simple groups. Our proof relies on a result of Meierfrankenfeld and Stroth [MS] on quadratic four-groups and on the Baumann-Glauberman-Niles theorem, for which Stellmacher [St2] has given an amalgam-theoretic proof. Apart from those results, our proof is essentially self-contained. For John Thompson Supported in part by NSF grant #DMS 89-03124, by DIMACS, an NSF Science and Technology Center, funded under contract STC-88-09648, and by NSA grant #MDA-904-91-H-0043. Prof. Gorenstein died on August 26, 1992.     

Strict Betweennesses Induced by Posets as well as by Graphs

For a finite poset P = (V, ≤ ), let _s(P){\cal B}_s(P) consist of all triples (x,y,z) ∈ V ³ such that either x < y < z or z < y < x. Similarly, for every finite, simple, and undirected graph G = (V,E), let B_s(G){\cal B}_s(G) consist of all triples (x,y,z) ∈ V ³ such that y is an internal vertex on an induced path in G between x and z. The ternary relations B_s(P){\cal B}_s(P) and B_s(G){\cal B}_s(G) are well-known examples of so-called strict betweennesses. We characterize the pairs (P,G) of posets P and graphs G on the same ground set V which induce the same strict betweenness relation B_s(P)=B_s(G){\cal B}_s(P)={\cal B}_s(G).     

Topological *-algebras withC*-enveloping algebras II

UniversalC*-algebrasC*(A) exist for certain topological *-algebras called algebras with aC*-enveloping algebra. A Frechet *-algebraA has aC*-enveloping algebra if and only if every operator representation ofA mapsA into bounded operators. This is proved by showing that every unbounded operator representation π, continuous in the uniform topology, of a topological *-algebraA, which is an inverse limit of Banach *-algebras, is a direct sum of bounded operator representations, thereby factoring through the enveloping pro-C*-algebraE(A) ofA. Given aC*-dynamical system (G,A,α), any topological *-algebraB containingC _c (G,A) as a dense *-subalgebra and contained in the crossed productC*-algebraC*(G,A,α) satisfiesE(B) =C*(G,A,α). IfG = ℝ, ifB is an α-invariant dense Frechet *-subalgebra ofA such thatE(B) =A, and if the action α onB ism-tempered, smooth and by continuous *-automorphisms: then the smooth Schwartz crossed productS(ℝ,B,α) satisfiesE(S(ℝ,B,α)) =C*(ℝ,A,α). WhenG is a Lie group, theC ^∞-elementsC ^∞(A), the analytic elementsC ^ω(A) as well as the entire analytic elementsC ^є(A) carry natural topologies making them algebras with aC*-enveloping algebra. Given a non-unitalC*-algebraA, an inductive system of idealsI _α is constructed satisfyingA =C*-ind limI _α; and the locally convex inductive limit ind limI _α is anm-convex algebra with theC*-enveloping algebraA and containing the Pedersen idealK _a ofA. Given generatorsG with weakly Banach admissible relationsR, we construct universal topological *-algebraA(G, R) and show that it has aC*-enveloping algebra if and only if (G, R) isC*-admissible.     

On Groups with Finite Involution and Locally Finite 2-Isolated Subgroup of Even Period

A proper subgroup H of a group G is said to be strongly isolated if it contains the centralizer of any nonidentity element of H and 2-isolated if the conditions >C _G(g) H 1 and 2(C_G(g)) imply that C_G(g)H. An involution i in a group G is said to be finite if |ii ^g| < (for any g G). In the paper we study a group G with finite involution i and with a 2-isolated locally finite subgroup H containing an involution. It is proved that at least one of the following assertions holds:1) all 2-elements of the group G belong to H;2) (G,H) is a Frobenius pair, H coincides with the centralizer of the only involution in H, and all involutions in G are conjugate;3) G=FFC_G(i) is a locally finite Frobenius group with Abelian kernel F;4) H=V D is a Frobenius group with locally cyclic noninvariant factor D and a strongly isolated kernel V, U=O₂(V) is a Sylow 2-subgroup of the group G, and G is a Z-group of permutations of the set =U ^g g G.     

Sums of corestrictions of cyclic algebras

By a cyclic layer of a finite Galois extension,E/K, of fields one means a cyclic extension,L/F, of fields whereE⊇L⊇F⊇K. LetC(E/K) denote the subgroup of the relative Brauer group, Br(E/K), generated by the various subgroups cor(Br(L/F)) asL/F ranges over all cyclic layers ofE/K and where cor denotes the corestriction map into Br(E/K). We show that forK global, [Br(E/K) :C(E/K)]<∞ and we produce examples whereC(E/K)≠Br(E/K). In memory of S.A. Amitsur, our teacher, friend, collaborator, and inspiration. Supported in part by NSA Grant No. MDA904-95-H-1054. Supported in part by NSA Grant No. MDA904-95-H-1022.     

On Some Three-Color Ramsey Numbers

 In this paper we study three-color Ramsey numbers. Let K _i,j denote a complete i by j bipartite graph. We shall show that (i) for any connected graphs G ₁, G ₂ and G ₃, if r(G ₁, G ₂)≥s(G ₃), then r(G ₁, G ₂, G ₃)≥(r(G ₁, G ₂)−1)(χ(G ₃)−1)+s(G ₃), where s(G ₃) is the chromatic surplus of G ₃; (ii) (k+m−2)(n−1)+1≤r(K _1,k , K _1,m , K _n )≤ (k+m−1)(n−1)+1, and if k or m is odd, the second inequality becomes an equality; (iii) for any fixed m≥k≥2, there is a constant c such that r(K _k,m , K _k,m , K _n )≤c(n/logn), and r(C _2m , C _2m , K _n )≤c(n/logn) ^m/(m−1) for sufficiently large n. Received: July 25, 2000 Final version received: July 30, 2002 RID="*" ID="*" Partially supported by RGC, Hong Kong; FRG, Hong Kong Baptist University; and by NSFC, the scientific foundations of education ministry of China, and the foundations of Jiangsu Province Acknowledgments. The authors are grateful to the referee for his valuable comments. AMS 2000 MSC: 05C55     

Typical Frobenius coverings

A covering p from a Cayley graph Cay（G, X） onto another Cay（H, Y） is called typical Frobenius if G is a Frobenius group with H as a Frobenius complement and the map p ： G →H is a group epimorphism. In this paper, we emphasize on the typical Frobenius coverings of Cay（H, Y）. We show that any typical Frobenius covering Cay（G, X） of Cay（H, Y） can be derived from an epimorphism /from G to H which is determined by an automorphism f of H. If Cay（G, X1） and Cay（G, X2） are two isomorphic typical Frobenius coverings under a graph isomorphism Ф, some properties satisfied by Фare given.     

On the fundamental group of real toric varieties

LetX (Δ) be the real toric variety associated to a smooth fan Δ. The main purpose of this article is: (i) to determine the fundamental group and the universal cover ofX (Δ), (ii) to give necessary and sufficient conditions on Δ under which π₁(X(Δ)) is abelian, (iii) to give necessary and sufficient conditions on Δ under whichX(Δ) is aspherical, and when Δ is complete, (iv) to give necessary and sufficient conditions forC _Δ to be aK (π, 1) space whereC _Δ is the complement of a real subspace arrangement associated to Δ.     

Affinely equivalent complete flat manifolds

Let E _Aff(Γ,G, m) be the set of affine equivalence classes of m-dimensional complete flat manifolds with a fixed fundamental group Γ and a fixed holonomy group G. Let n be the dimension of a closed flat manifold whose fundamental group is isomorphic to Γ. We describe E _Aff(Γ,G, m) in terms of equivalence classes of pairs (ε, ρ), consisting of epimorphisms of Γ onto G and representations of G in ℝ ^m-n . As an application we give some estimates of card E _Aff(Γ,G, m).     

On the computational complexity of ordered subgraph recognition

Let (G, <) be a finite graph G with a linearly ordered vertex set V. We consider the decision problem (G, <)ORD to have as an instance an (unordered) graph Γ and as a question whether there exists a linear order ≤ on V(Γ) and an order preserving graph isomorphism of (G, <) onto an induced subgraph of Γ. Several familiar classes of graph are characterized as the yes-instances of (G, < )ORD for appropriate choices of (G, <). Here the complexity of (G, <)ORD is investigated. We conjecture that for any 2-connected graph G, G ≠ K_k, (G, <)ORD is NP-complete. This is verified for almost all 2-connected graphs. Several related problems are formulated and discussed. © 1995 John Wiley & Sons, Inc.     

The McKay conjecture for exceptional groups and odd primes

Let G be a simply-connected simple algebraic group over an algebraically closed field of characteristic p with a Frobenius map F : G → G and G := G ^F , such that the root system is of exceptional type or G is a Suzuki group or Steinberg’s triality group. We show that all irreducible characters of C _G (S), the centraliser of S in G, extend to their inertia group in N _G (S), where S is any F-stable Sylow torus of (G, F). Together with the work in [16] this implies that the McKay conjecture is true for G and odd primes ℓ different from the defining characteristic. Moreover it shows important properties of the associated simple groups, which are relevant for the proof that the associated simple groups are good in the sense of Isaacs, Malle and Navarro, as defined in [14]. This research has been supported by the DFG-grant “Die Alperin-McKay-Vermutung für endliche Gruppen” and an Oberwolfach Leibniz fellowship.