Permutability of subgroups of G×H that are direct products of subgroups of the direct factors

If M and S are two subgroups of a group G, M and S permute if MS = SM. Furthermore, M is a permutable subgroup of G if M permutes with every subgroup of G. We give necessary and sufficient conditions for M, a subgroup of G, to permute with a subgroup of G 2 H given that G and H are finite groups. The main part of the paper involves the development of a characterization of permutable subgroups of G 2 H that are direct products of subgroups of the direct factors; that is, subgroups that are equal to A 2 B where A \leqq \leqq G and B \leqq \leqq H.     

The 2-Pebbling Property and a Conjecture of Graham's

The pebbling number of a graph G, f(G), is the least m such that, however m pebbles are placed on the vertices of G, we can move a pebble to any vertex by a sequence of moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. It is conjectured that for all graphs G and H, f(G 2H)hf(G)f(H).¶Let C_m and C_n be cycles. We prove that f(C_m 2C_n)hf(C_m) f(C_n) for all but a finite number of possible cases. We also prove that f(G2T)hf(G) f(T) when G has the 2-pebbling property and T is any tree.     

Auto-equivalences of derived categories acting on cohomology

Let A be a k-algebra which is projective as a k-module, let M be an A-module whose endomorphisms are given by multiplication by central elements of A, and let TrPic_k(A) be the group of standard self-equivalences of the derived category of bounded complexes of A-modules. Then we define an action of the stabilizer of M in TrPic_k(A) on the Ext-algebra of M. In case M is the trivial module for the group algebra kG = A, this defines an action on the cohomology ring of G which extends the well-known action of the automorphism group of G on the cohomology group.     

On a filtered multiplicative basis of group algebras

Let K be a field of characteristic p and G a nonabelian metacyclic finite p-group. We give an explicit list of all metacyclic p-groups G, such that the group algebra KG over a field of characteristic p has a filtered multiplicative K-basis. We also present an example of a non-metacyclic 2-group G, such that the group algebra KG over any field of characteristic 2 has a filtered multiplicative K-basis.     

The determination of the thermal properties of a heat conductor in a nonlinear heat conduction problem

This paper considers the inverse determination of the positive unknown thermal properties K(T), C(T) and the unknown temperature T(_{x, t}) in the nonlinear transient heat conduction equation. In addition to prescribed initial and/or boundary values, specified continuously differentiable temperature data T(x₀, t) with non-zero derivative at a single sensor location x = x₀ is given. When K(T) and C(T) obey a certain relationship which enables one to linearise exactly the nonlinear heat equation then their dependence upon T is obtained explicitly, whilst the unknown temperature T(x, t) is obtained implicitly and is then calculated numerically. Results are presented and discussed for infinite, semi-infinite and finite slabs.     

On self-normality and abnormality in the alternating group Ap

An old problem proposed by Huppert, Doerk and Hawkes motivates us to investigate the relationship between an abnormal subgroup and self-normalizing in non-solvable groups. A subgroup H of a group G is called second maximal if H is maximal in all maximal subgroups of G containing H. Our result is that if H is a second maximal subgroup of the alternating group A_p of prime degree, then H is abnormal in A_p if and only if H is self-normalizing.     

Group Connectivity of 3-Edge-Connected Chordal Graphs

Let A be a finite abelian group and G be a digraph. The boundary of a function f: E(G)ZA is a function ‘f: V(G)ZA given by ‘f(v)=~_{e leaving v}f(e)m~_{e entering v}f(e). The graph G is A-connected if for every b: V(G)ZA with ~_{v] V(G)} b(v)=0, there is a function f: E(G)ZA{0} such that ‘f=b. In [J. Combinatorial Theory, Ser. B 56 (1992) 165-182], Jaeger et al showed that every 3-edge-connected graph is A-connected, for every abelian group A with |A|̈́. It is conjectured that every 3-edge-connected graph is A-connected, for every abelian group A with |A|̓ and that every 5-edge-connected graph is A-connected, for every abelian group A with |A|́.¶ In this note, we investigate the group connectivity of 3-edge-connected chordal graphs and characterize 3-edge-connected chordal graphs that are A-connected for every finite abelian group A with |A|́.     

The VT-operator Semigroup for Two Kinds of Regular Semigroups

Let S be a regular semigroup and Con S the congruence lattice of S. For every rho element of Con S there exists a greatest congruence rhoV [smallest congruence rhov] on S such that the idempotent (rhoV/rho)-classes [(rho/rhov)-classes] are rectangular bands, and a greatest congruence rhoT [smallest congruence rhot] on S such that the idempotent (rhoT/rho-classes [(rho/rhot-classes] are groups. The subsemigroup of the transformation semigroup on Con S generated by the transformation rho → rhoV, rho → rhov, rho → rhoT, and rho → rhot, rho element of Con S, is investigated for orthodox semigroups and cryptogroups. It is shown that in this case this so-called Vt-operator semigroup Omega(S) contains 17 elements at most. A 17-element Vt-operator semigroup Omega(F) is realized for some regular orthogroup F.     

Independent Trees in Planar Graphs Independent trees

We prove the following statement: Let G be a finite k-connected undirected planar graph and s be a vertex of G. Then there exist k spanning trees T₁,…,T_k in G such that for each vertex xps of G, the k paths from x to s in T₁,…,T_k are pairwise openly disjoint.     

Helly-type theorems for appropriate colorings of visibility sets

Summary. For positive integers q and n, think of P as the vertex set of a (qn + r)-gon, 0 ￡ r ￡ q - 1 0 \leq r \leq q - 1 . For 1 ￡ i ￡ qn + r 1 \leq i \leq qn + r , define V(i) to be a set of q consecutive points of P, starting at p(i), and let S be a subset of {V(i) : 1 ￡ i ￡ qn + r } \lbrace V(i) : 1 \leq i \leq qn + r \rbrace . A q-coloring of P = P(q) such that each member of S contains all q colors is called appropriate for S, and when 1 ￡ j ￡ q 1 \leq j \leq q , the definition may be extended to suitable subsets P(j) of P. If for every 1 ￡ j ￡ q 1 \leq j \leq q and every corresponding P(j), P(j) has a j-coloring appropriate for S, then we say P = P(q) has all colorings appropriate for S. With this terminology, the following Helly-type result is established: Set P = P(q) has all colorings appropriate for S if and only if for every (2n + 1)-member subset T of S, P has all colorings appropriate for T. The number 2n + 1 is best possible for every r 3 1 r \geq 1 . Intermediate results for q-colorings are obtained as well.     

Cyclic codes and the Frobenius automorphism

Using the theory of cyclic codes the following problem of K. Burde' on characterizing finite fields GF (qⁿ) is solved:¶ Consider GF (qⁿ) as a vector space over GF (q). For which GF (qⁿ) exists for any k = 0, . . . ,n exactly one subspace C of dimension k and which is invariant under the Frobenius automorphism?     

Quasi-polynomials associated with Bass and Betti numbers of filtered modules

It is shown that if M is a finite module on a local noetherian ring A which is filtered by an f-good filtration F \Phi = (M_n) where f is a noetherian filtration on A, then the i-th Betti and the i-th Bass numbers of the modules (M_n) and (M / M_n) define quasi-polynomial functions whose period does not depend on i but only of the Rees ring of f. It is proved that the projective and injective dimension of the modules M / M_n are perodic for large n. In the particular case where f is a good filtration or a strongly A P filtration it is shown that the projective and injective dimension as well as the depth stabilize. As an application, using a result proved by Brodmann, we give an upper bound of the analytic spread of¶ f = (I_n) in terms of the limes inferior of depth (A / I_n).     

Power residues on Abelian varieties

Given a prime l and an elliptic curve E defined over a number field k, we show that a non-zero point P] E(k) lies in lE(k) if and only if P lies in lE(k)(mod ) for almost all finite primes  of k. We give conditions on l under which analogous results hold for Abelian varieties and with one point replaced by a finite number of points. We also construct examples to show that these conditions are essential.     

Regular orbits in powers of permutation representations

Let (Q,G) be a faithful permutation representation of a finite group G. Suppose that the G-set Q has t distinct non-zero marks. In a permutation representation analogue of a theorem of Brauer on linear representations, it is shown that the direct power (Q,G)^t of (Q,G) contains a regular orbit. As a corollary, the probability that a random element of Q^r lies in a regular orbit of (Q,G)^r is shown to tend to 1 exponentially fast as r tends to \infin\infin. Further, knowledge of the rate of convergence is equivalent to knowledge of the second largest value of the character of the linear permutation representation.     

On Groups Associated with Transformation Semigroups

For a semigroup S of transformations (total or partial) of a finite n-element set X_n, denote by G_S the group of all the permutations h of X_n that preserve S under conjugation. It is shown that, unless S contains certain nilpotents and has a very restricted form, the alternating group Alt_n may not serve as G_S, so that Alt_n ⊆ G_S implies that G_S=S_n, and S is an S_n-normal semigroup.     

On the binomial arithmetical rank

The binomial arithmetical rank of a binomial ideal I is the smallest integer s for which there exist binomials f₁,..., f_s in I such that rad (I) = rad (f₁,..., f_s). We completely determine the binomial arithmetical rank for the ideals of monomial curves in P_KⁿP_K^n. In particular we prove that, if the characteristic of the field K is zero, then bar (I(C)) = n - 1 if C is complete intersection, otherwise bar (I(C)) = n. While it is known that if the characteristic of the field K is positive, then bar (I(C)) = n - 1 always.     

Invariants of free Lie algebras

Let k be a principal ideal domain with identity and characteristic zero. For a positive integer n, with n \geqq 2n \geqq 2, let H(n) be the group of all n x n matrices having determinant ±1\pm 1. Further, we write SL(n) for the special linear group. Let L be a free Lie algebra (over k) of finite rank n. We prove that the algebra of invariants L^B(n) of B(n), with B(n) ? { H(n), SL(n)}B(n) \in \{ H(n), {\rm SL}(n)\} , is not a finitely generated free Lie algebra. Let us assume that k is a field of characteristic zero and let áSem(n) ?\langle {\rm Sem}(n) \rangle be the Lie subalgebra of L generated by the semi-invariants (or Lie invariants) Sem(n). We prove that áSem(n) ?\langle {\rm Sem}(n) \rangle is not a finitely generated free Lie algebra which gives a positive answer to a question posed by M. Burrow [4].     

Bireflectionality of orthogonal and symplectic groups of characteristic 2

Let V be a finite dimensional vector space over a field K of characteristic 2. Let O(V) be the orthogonal group defined by a nondegenerate quadratic form. Then every element in O(V) is a product of two elements of order 2, unless all nonsingular subspaces of V are at most 2-dimensional. If V is a nonsingular symplectic space, then every element in the symplectic group Sp (V) is a product of two elements of order 2, except if dim V = 2 and |K| = 2.     

The dimension monoid of a lattice

We introduce the dimension monoid of a lattice L, denoted by Dim L. The monoid Dim L is commutative and conical, the latter meaning that the sum of any two nonzero elements is nonzero. Furthermore, Dim L is given along with the dimension map, D\Delta from L2L to Dim L, which has the intuitive meaning of a distance function. The maximal semilattice quotient of Dim L is isomorphic to the semilattice Con_c L of compact congruences of L; hence Dim L is a precursor of the congruence lattice of L. Here are some additional features of this construction: ¶¶ (1) Our dimension theory provides a generalization to all lattices of the von Neumann dimension theory of continuous geometries. In particular, if L is an irreducible continuous geometry, then Dim L is either isomorphic to \Bbb Z⁺\Bbb Z^+ or to \Bbb R⁺\Bbb R^+.¶ (2) If L has no infinite bounded chains, then Dim L embeds (as an ordered monoid) into a power of \Bbb Z⁺è{￥}.{\Bbb Z}^{+}\cup \{\infty\}.¶ (3) If L is modular or if L has no infinite bounded chains, then Dim L is a refinement monoid.¶ (4) If L is a simple geometric lattice, then Dim L is isomorphic to \Bbb Z⁺\Bbb Z^+, if L is modular, and to the two-element semilattice, otherwise.¶ (5) If L is an à₀\aleph_0-meet-continuous complemented modular lattice, then both Dim L and the dimension function D\Delta satisfy (countable) completeness properties.¶¶ If R is a von Neumann regular ring and if L is the lattice of principal right ideals of the matrix ring M₂ (R), then Dim L is isomorphic to the monoid V (R) of isomorphism classes of finitely generated projective right R-modules. Hence the dimension theory of lattices provides a wide lattice-theoretical generalization of nonstable K-theory of regular rings.