Maximal Holonomy of Almost Bieberbach Groups for Heis5

Let Heis _2n+1 be the Heisenberg group of dimension 2n + 1 and M an infra-nilmanifold with Heis _2n+1-geometry. The fundamental group of M contains a cocompact lattice of Heis _2n+1 with index bounded above by a universal constant I _n+1, i.e., I _n+1 is the maximal order of the holonomy groups. We prove that I ₃ = 24. As an application we give an estimate for the volumes of finite volume non-compact complex hyperbolic 3-manifolds.     

A 3-local Characterization for M 11

We identify M ₁₁ by the structure of the normalizer of its Sylow 3-subgroup.     

M 0 Measures for the Walsh System

We show that the spaces M ₀(T) and M₀(D)M_{0}(\mathcal {D}) cannot be identified, D\mathcal {D} being the Cantor group ∏ _n=1^∞{±}. Our motivation is guided by the role M ₀ measures play in uniform distribution problems, via the Davenport, Erdòs, LeVeque theorem.     

The M3[D] construction and n-modularity

In 1968, Schmidt introduced the M ₃[D] construction, an extension of the five-element modular nondistributive lattice M ₃ by a bounded distributive lattice D, defined as the lattice of all triples satisfying . The lattice M ₃[D] is a modular congruence-preserving extension of D.? In this paper, we investigate this construction for an arbitrary lattice L. For every n > 0, we exhibit an identity such that is modularity and is properly weaker than . Let M _n denote the variety defined by , the variety of n-modular lattices. If L is n-modular, then M ₃[L] is a lattice, in fact, a congruence-preserving extension of L; we also prove that, in this case, Id M ₃[L] M ₃[Id L]. ? We provide an example of a lattice L such that M ₃[L] is not a lattice. This example also provides a negative solution to a problem of Quackenbush: Is the tensor product of two lattices A and B with zero always a lattice. We complement this result by generalizing the M ₃[L] construction to an M ₄[L] construction. This yields, in particular, a bounded modular lattice L such that M ₄ L is not a lattice, thus providing a negative solution to Quackenbush’s problem in the variety M of modular lattices.? Finally, we sharpen a result of Dilworth: Every finite distributive lattice can be represented as the congruence lattice of a finite 3-modular lattice. We do this by verifying that a construction of Gr?tzer, Lakser, and Schmidt yields a 3-modular lattice. Received May 26, 1998; accepted in final form October 7, 1998.     

Integral Group Ring of the Mathieu Simple Group M 23

We investigate the classical Zassenhaus conjecture for the unit group of the integral group ring of Mathieu simple group M ₂₃ using the Luthar–Passi method. This work is a continuation of the research that we carried out for Mathieu groups M ₁₁ and M ₁₂. As a consequence, for this group we confirm Kimmerle's conjecture on prime graphs.     

On a Rank Five Geometry of Meixner for the Mathieu Group M12

We construct a rank five residually connected and firm geometry on which the Mathieu group M ₁₂ acts flag-transitively and residually weakly primitively (RWPRI). The group M ₁₂ is the group of automorphisms of and Aut(M ₁₂) is the correlation group of , in particular is self-dual. The diagram of is the following. Moreover satisfies the conditions (IP)₂ and (2T)₁. As a corollary, we obtain that the (RWPRI+(IP)₂)-rank of M ₁₂ is 5.     

Chow group of 1-cycles on the moduli space of vector bundles of rank 2 over a curve

Let X be a non-singular complex projective curve of genus ≥3. Choose a point x ∈ X. Let M_x be the moduli space of stable bundles of rank 2 with determinant We prove that the Chow group CH^Q₁(M_x) of 1-cycles on M_x with rational coefficients is isomorphic to CH^Q₀(X). By studying the rational curves on M_x, it is not difficult to see that there exits a natural homomorphism CH₀(J)→CH₁(M_x) where J denotes the Jacobian of X. The crucial point is to show that this homomorphism induces a homomorphism CH₀(X)→CH₁(M_x), namely, to go from the infinite dimensional object CH₀(J) to the finite dimensional object CH₀(X). This is proved by relating the degeneration of Hecke curves on M_x to the second term I^*2 of Bloch's filtration on CH₀(J). Insong Choe was supported by KOSEF (R01-2003-000-11634-0).     

ON X-EXTENDING AND X-CONTINUOUS MODULES

Let X be a left R-module. We characterize when the direct sum of two X-extending modules is X-extending via essential injectivity and pseudo injectivity of modules. As a corollary, we show that if extending modules M ₁ and M ₂ are relatively essentially injective and M ₁ is pseudo-M ₂-injective (or M ₂ is pseudo-M ₁-injective) then M ₁ ⊕ M ₂ is extending. Also we characterize when the direct sum of two CESS-modules is CESS. Some characterizations of almost Noetherian rings are also given by relative (quasi-) continuity of left R-modules.     

Polynomial Identities of M2,1(G)

We describe the multilinear identities of the superalgebra M _{2, 1}(G) of matrices over the Grassmann algebra, in the minimal possible degree, which is 9.     

A New Characterization of A 10 by its Noncommuting Graph

Let G be a nonabelian group and associate a noncommuting graph ?(G) with G as follows: The vertex set of ?(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In 1987, Professor J. G. Thompson gave the following conjecture.

Thompson's Conjecture. If G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying N(G) = N(M), then G ? M, where N(G):={n ∈ ? | G has a conjugacy class of size n}.

In 2006, A. Abdollahi, S. Akbari, and H. R. Maimani put forward a conjecture (AAM's conjecture) in Abdollahi et al. (2006) as follows.

AAM's Conjecture. Let M be a finite nonabelian simple group and G a group such that ?(G) ? ? (M). Then G ? M.

In this short article we prove that if G is a finite group with ?(G) ? ? (A ₁₀), then G ? A ₁₀, where A ₁₀ is the alternating group of degree 10.     

On v-closed manis valution rings

ABSTRACT

The graded Lie algebra L associated to the Nottingham group with respect to its natural filtration is known to be a loop algebra of the first Witt algebra W ₁ . The fact that the Schur multiplier of W ₁ , in characteristic p > 3, is one-dimensional implies that L is not finitely presented. Consider the universal covering ? ₁ of W ₁ and the corresponding loop algebra M of ? ₁ . In this paper we prove that M itself is finitely presented for p > 3. In characteristic p >  11 the algebra M turns out to be presented by two relations.     

Some Extensions of PSL(2, p 2) are Uniquely Determined by Their Complex Group Algebras

In Tong-Viet's, 2012 work, the following question arose: Question. Which groups can be uniquely determined by the structure of their complex group algebras?

It is proved here that some simple groups of Lie type are determined by the structure of their complex group algebras. Let p be an odd prime number and S = PSL(2, p ²). In this paper, we prove that, if M is a finite group such that S < M < Aut(S), M = ?₂ × PSL(2, p ²) or M = SL(2, p ²), then M is uniquely determined by its order and some information about its character degrees. Let X ₁(G) be the set of all irreducible complex character degrees of G counting multiplicities. As a consequence of our results, we prove that, if G is a finite group such that X ₁(G) = X ₁(M), then G ? M. This implies that M is uniquely determined by the structure of its complex group algebra.     

On the Finite Group with the Index of Maximal Subgroups of the Monster

Abstract

We study the problem concerning the influence of the index of maximal subgroup or the degree of primitive permutation representation of the finite simple groups on the structure of a group. Let G be a finite group and s be the index of maximal subgroup of the Monster M. In this paper, we prove that there exists an epimorphism from G to M or A _s if G has the primitive permutation representation of degree s, and as a consequence we prove that the Monster is determined by every s.     

On Generation of Sporadic Simple Groups by Three Involutions Two of Which Commute

We prove the following result: Let G be one of the 26 sporadic simple groups. The group G cannot be generated by three involutions two of which commute if and only if G is isomorphic to M ₁₁, M ₂₂, M ₂₃, or M ^cL .     

Finite Groups with S-permutable n-maximal Subgroups

Let G be a finite group and M _n (G) be the set of n-maximal subgroups of G, where n is an arbitrary given positive integer. Suppose that M _n (G) contains a nonidentity member and all members in M _n (G) are S-permutable in G. Then any of of the following conditions guarantees the supersolvability of G: (1) M _n (G) contains a nonidentity member whose order is not a prime; (2) all nonidentity members in M _n (G) are of prime order, and all cyclic members in M _n?1(G) of order 4 are S-permutable in G.     

{\sl M}-classification of Regular Semigroups

Abstract. On any regular semigroup S, the least group congruence σ, the greatest idempotent pure congruence τ and the least band congruence β are used to give the M -classification of regular semigroups as follows. These congruences generate a sublattice Λ of the congruence lattice C (S) of S. We consider the triples (Λ,K,T), where K and T are the restrictions of the K - and T -relations on {C (S) to Λ. Such triples are characterized abstractly and form the objects of a category M whose morphisms are surjective T -preserving homomorphisms subject to a mild condition. The class of regular semigroups is made into a category M whose morphisms are fairly restricted homomorphisms. The main result of the paper is the existence of a representative functor from M to M. Several properties of the classification of regular semigroups induced by this functor are established.     

On the Order Dimension of Outerplanar Maps

Schnyder characterized planar graphs in terms of order dimension. Brightwell and Trotter proved that the dimension of the vertex-edge-face poset P _M of a planar map M is at most four. In this paper we investigate cases where dim(P _M ) ≤ 3 and also where dim(Q _M ) ≤ 3; here Q _M denotes the vertex-face poset of M. We show:

Commensurability of Hyperbolic Manifolds with Geodesic Boundary

Suppose , let M ₁, M ₂ be n-dimensional connected complete finite-volume hyperbolic manifolds with nonempty geodesic boundary, and suppose that π₁ (M ₁) is quasi-isometric to π₁ (M ₂) (with respect to the word metric). Also suppose that if n=3, then ∂M ₁ and ∂M ₂ are compact. We show that M ₁ is commensurable with M ₂. Moreover, we show that there exist homotopically equivalent hyperbolic 3-manifolds with non-compact geodesic boundary which are not commensurable with each other. We also prove that if M is as M ₁ above and G is a finitely generated group which is quasi-isometric to π₁ (M), then there exists a hyperbolic manifold with geodesic boundary M′ with the following properties: M′ is commensurable with M, and G is a finite extension of a group which contains π₁ (M′) as a finite-index subgroupMathematics Subject Classification (2000). Primary: 20F65; secondary: 30C65, 57N16     

Cobordism of immersions of surfaces¶in non-orientable 3-manifolds

Some properties of non-orientable 3-manifolds are shown. In particular, for a connected, non-orientable 3-manifold M, the group of cobordism clases of immersions of surfaces in M is isomorphic to a group structure on the set H ₂(M,Z/2Z)×H ₁(M,Z/2Z)×Z/2Z. Received: 8 June 2000 / Revised version: 2 October 2000