Mutually permutable subgroups and group classes

We consider the product AB of two finite mutually permutable subgroups A, B and find some subnormal subgroups of the product. This leads to local and otherwise generalized statements about products of supersolvable groups.Received: 19 May 2004     

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.     

The McKay-Thompson Series of Some Elements of Order 3,5,and 7 in the Monster

Abstract

In this article,we explicitly calculate the McKay-Thompson series corresponding to 3A,7A,3B,5B and 15C elements of the Monster simple group by using the framed VOA structure of the Moonshine VOA and automorphisms which permute 48 mutually orthogonal rational conformal vectors with central charge 1/2.     

Almost isomorphism of Abelian groups and determinability of Abelian groups by their subgroups

An Abelian group A is called correct if for any Abelian group B isomorphisms A ≅ B′ and B ≅ A′, where A′ and B′ are subgroups of the groups A and B, respectively, imply the isomorphism A ≅ B. We say that a group A is determined by its subgroups (its proper subgroups) if for any group B the existence of a bijection between the sets of all subgroups (all proper subgroups) of groups A and B such that corresponding subgroups are isomorphic implies A ≅ B. In this paper, connections between the correctness of Abelian groups and their determinability by their subgroups (their proper subgroups) are established. Certain criteria of determinability of direct sums of cyclic groups by their subgroups and their proper subgroups, as well as a criterion of correctness of such groups, are obtained. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 21–36, 2003.     

On a graph related to permutability in finite groups

For a finite group G we define the graph Γ(G) to be the graph whose vertices are the conjugacy classes of cyclic subgroups of G and two conjugacy classes ${\mathcal {A}, \mathcal {B}}For a finite group G we define the graph Γ(G) to be the graph whose vertices are the conjugacy classes of cyclic subgroups of G and two conjugacy classes A, B{\mathcal {A}, \mathcal {B}} are joined by an edge if for some A ? A, B ? B A{A \in \mathcal {A},\, B \in \mathcal {B}\, A} and B permute. We characterise those groups G for which Γ(G) is complete.     

Local commensurability graphs of solvable groups

The commensurability index between two subgroups A,B of a group G is [A:A∩B][B:A∩B]. This gives a notion of distance among finite index subgroups of G, which is encoded in the p-local commensurability graphs of G. We show that for any metabelian group, any component of the p-local commensurabilty graph of G has diameter bounded above by 4. However, no universal upper bound on diameters of components exists for the class of finite solvable groups. In the appendix we give a complete classification of components for upper triangular matrix groups in GL(2,𝔽_q).     

Preserving Zeros of xy and xy*

We describe surjective additive maps θ: A → B which preserve zero products, where A is a ring with a nontrivial idempotent and B is a prime ring. We also characterize surjective additive maps θ: A → B such that for all x, y ∈ A we have θ(x)θ(y)* = 0 if and only if xy* = 0. Here A is a unital prime ring with involution that contains a nontrivial idempotent and B is a prime ring with involution.     

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 ?  ?.     

A Lower Bound for {a+b:aA, bB, and P(a,b)≠0}

Let A and B be two finite subsets of a field . In this paper, we provide a non-trivial lower bound for {a+b:aA, bB, and P(a,b)≠0} where P(x,y) [x,y].     

The structure of the 3-separations of 3-connected matroids

Tutte defined a k-separation of a matroid M to be a partition (A,B) of the ground set of M such that |A|,|B|k and r(A)+r(B)−r(M)<k. If, for all m<n, the matroid M has no m-separations, then M is n-connected. Earlier, Whitney showed that (A,B) is a 1-separation of M if and only if A is a union of 2-connected components of M. When M is 2-connected, Cunningham and Edmonds gave a tree decomposition of M that displays all of its 2-separations. When M is 3-connected, this paper describes a tree decomposition of M that displays, up to a certain natural equivalence, all non-trivial 3-separations of M.     

Extending Tilting Modules to One-Point Extensions by Projectives

Let k be an algebraically closed field, B be a finite dimensional k-algebra and A be the one-point extension of B by the projective B-module P ₀. We compare the posets 𝒯 _A and 𝒯 _B of tilting A-modules and B-modules, respectively. We prove that the restriction and the extension functors define morphisms of posets r: 𝒯 _A  → 𝒯 _B and e: 𝒯 _B  → 𝒯 _A such that re = id. Moreover, e induces a full embedding of the quiver of 𝒯 _B into that of 𝒯 _A , whose image is closed under successors, and mapping distinct connected components of the first into distinct connected components of the second.     

2-uniform congruences in majority algebras and a closure operator

A ternary term m(x, y, z) of an algebra is called a majority term if the algebra satisfies the identities m(x, x, y) = x, m(x, y, x) = x and m(y, x, x) = x. A congruence α of a finite algebra is called uniform if all of its blocks (i.e., classes) have the same number of elements. In particular, if all the α-blocks are two-element then α is said to be a 2-uniform congruence. If all congruences of A are uniform then A is said to be a uniform algebra. Answering a problem raised by Gr?tzer, Quackenbush and Schmidt [2], Kaarli [3] has recently proved that uniform finite lattices are congruence permutable. In connection with Kaarli’s result, our main theorem states that for every finite algebra A with a majority term any two 2-uniform congruences of A permute. Examples show that we can say neither “algebra” instead of “algebra with a majority term”, nor “3-uniform” instead of “2-uniform”. Given two nonempty sets A and B, each relation gives rise to a pair of closure operators, which are called the Galois closures on A and B induced by ρ. Galois closures play an important role in many parts of algebra, and they play the main role in formal concept analysis founded by Wille [4]. In order to prove our main theorem, we introduce a pair of smaller closure operators induced by ρ. These closure operators will hopefully find further applications in the future. Dedicated to the memory of Kazimierz Głazek Presented by E. T. Schmidt. Received November 29, 2005; accepted in final form May 23, 2006. This research was partially supported by the NFSR of Hungary (OTKA), grant no. T049433 and T037877.     

Noncommutative algebras related with Schubert calculus on Coxeter groups

For any finite Coxeter system (W,S) we construct a certain noncommutative algebra, the so-called bracket algebra, together with a family of commuting elements, the so-called Dunkl elements. The Dunkl elements conjecturally generate an algebra which is canonically isomorphic to the coinvariant algebra of the Coxeter group W. We prove this conjecture for classical Coxeter groups and I₂(m). We define a “quantization” and a multiparameter deformation of our construction and show that for Lie groups of classical type and G₂, the algebra generated by Dunkl’s elements in the quantized bracket algebra is canonically isomorphic to the small quantum cohomology ring of the corresponding flag variety, as described by B. Kim. For crystallographic Coxeter systems we define the so-called quantum Bruhat representation of the corresponding bracket algebra. We study in more detail the structure of the relations in B_n-, D_n- and G₂-bracket algebras, and as an application, discover a Pieri-type formula in the B_n-bracket algebra. As a corollary, we obtain a Pieri-type formula for multiplication of an arbitrary B_n-Schubert class by some special ones. Our Pieri-type formula is a generalization of Pieri’s formulas obtained by A. Lascoux and M.-P. Schützenberger for flag varieties of type A. We also introduce a super-version of the bracket algebra together with a family of pairwise anticommutative elements, the so-called flat connections with constant coefficients, which describes “a noncommutative differential geometry on a finite Coxeter group” in the sense of S. Majid.     

Non-Nilpotent Graph of a Group

We associate a graph 𝒩 _G with a group G (called the non-nilpotent graph of G) as follows: take G as the vertex set and two vertices are adjacent if they generate a non-nilpotent subgroup. In this article, we study the graph theoretical properties of 𝒩 _G and its induced subgraph on G \ nil(G), where nil(G) = {x ∈ G | ? x, y ? is nilpotent for all y ∈ G}. For any finite group G, we prove that 𝒩 _G has either |Z*(G)| or |Z*(G)| +1 connected components, where Z*(G) is the hypercenter of G. We give a new characterization for finite nilpotent groups in terms of the non-nilpotent graph. In fact, we prove that a finite group G is nilpotent if and only if the set of vertex degrees of 𝒩 _G has at most two elements.     

On the groups SL2(ℤ[x]) and SL2(k[x,y])

This paper studies free quotients of the groups SL₂(ℤ[x]) and SL₂(k[x, y]),k a finite field. These quotients give information about the relation of the above groups to their subgroups generated by elementary or unipotent elements.     

Connectivity of intersection graphs of finite groups

The intersection graph of a group G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of G, and there is an edge between two distinct vertices H and K if and only if H∩K≠1 where 1 denotes the trivial subgroup of G. In this paper, we classify finite solvable groups whose intersection graphs are not 2-connected and finite nilpotent groups whose intersection graphs are not 3-connected. Our methods are elementary.     

The Yoneda Algebra of a Graded Ore Extension

Let A be a connected-graded algebra with trivial module 𝕜, and let B be a graded Ore extension of A. We relate the structure of the Yoneda algebra E(A): = Ext _A (𝕜, 𝕜) to E(B). Cassidy and Shelton have shown that when A satisfies their 𝒦₂ property, B will also be 𝒦₂. We prove the converse of this result.     

Finite Groups with Few TI-Subgroups

A subgroup A of a finite group G is called a TI-subgroup if either A ∩ A ^x  = 1 or A ∩ A ^x  = A holds for all x ∈ G. In this paper, finite group all of whose meta-cyclic subgroups are TI-subgroups are classified completely. In particular, such groups are solvable.     

The quasi-invariance property for the Gamma kernel determinantal measure

The Gamma kernel is a projection kernel of the form (A(x)B(y)−B(x)A(y))/(x−y), where A and B are certain functions on the one-dimensional lattice expressed through Euler's Γ-function. The Gamma kernel depends on two continuous parameters; its principal minors serve as the correlation functions of a determinantal probability measure P defined on the space of infinite point configurations on the lattice. As was shown earlier [A. Borodin, G. Olshanski, Adv. Math. 194 (2005) 141–202, arXiv:math-ph/0305043], P describes the asymptotics of certain ensembles of random partitions in a limit regime.Theorem: The determinantal measure P is quasi-invariant with respect to finitary permutations of the nodes of the lattice.This result is motivated by an application to a model of infinite particle stochastic dynamics.     

On Influence of Contranormal Subgroups on the Structure of Infinite Groups

Following Rose, a subgroup H of a group G is called contranormal, if G = H ^G . In certain sense, contranormal subgroups are antipodes to subnormal subgroups. It is well known that a finite group is nilpotent if and only if it has no proper contranormal subgroups. However, for the infinite groups this criterion is not valid. There are examples of non-nilpotent infinite groups whose subgroups are subnormal; in paricular, these groups have no contranormal subgroups. Nevertheless, for some classes of infinite groups, the absence of contranormal subgroups implies the nilpotency of the group. The current article is devoted to the search of such classes. Some new criteria of nilpotency in certain classes of infinite groups have been established.