Isomorphic toG Research Articles

Aα-Spectral Characterizations of Some Joins

LetGbe a graph withnvertices. For every realα∈0,1, writeAαGfor the matrixAαG=αDG+1−αAG, whereAGandDGdenote the adjacency matrix and the degree matrix ofG, respectively. The collection of eigenvalues ofAαGtogether with multiplicities are called theAα-spectrum ofG. A graphGis said to be determined by itsAα-spectrum if all graphs having the sameAα-spectrum asGare isomorphic toG. In this paper, we show that some joins are determined by theirAα-spectra forα∈0,1/2 or 1/2,1.

On the independence theorem of related structures for modular (arguesian) lattices

LetDbe a finite distributive lattice with more than one element and letGbe a finite group. We prove that there exists a modular (arguesian) lattice M such that the congruence lattice ofMis isomorphic toDand the automorphism group ofMis isomorphic toG.

Countable structures with a fixed group of automorphisms

We prove that, given a countable groupG, the set of countable structures (for a suitable languageL)U G whose automorphism group is isomorphic toG is a complete coanalytic set and ifG ≄H thenU G is Borel inseparable fromU H . We give also a model theoretic interpretation of this result. We prove, in contrast, that the set of countable structures forL whose automorphism group is isomorphic to ℤ p ℕ ,p a prime number, is Π 1 1 &σ 1 1 -complete.

On a characterization of finite vector bundles as vector bundles admitting a flat connection with finite monodromy group

We prove that a holomorphic vector bundle E E over a compact connected Kähler manifold admits a flat connection, with a finite group as its monodromy, if and only if there are two distinct polynomials f f and g g , with nonnegative integral coefficients, such that the vector bundle f ( E ) f(E) is isomorphic to g ( E ) g(E) . An analogous result is proved for vector bundles over connected smooth quasi-projective varieties, of arbitrary dimension, admitting a flat connection with finite monodromy group. When the base space is a connected projective variety, or a connected smooth quasi-projective curve, the above characterization of vector bundles admitting a flat connection with finite monodromy group was established by M. V. Nori.

Swan Modules and Realisable Classes for Kummer Extensions of Prime Degree

Letl>2 be a prime number. Let OFdenote a ring of algebraic integers of a number fieldFand letGbe the cyclic group of orderl. Consider the ring of integers OLas a locally free OK[G]-module whereL/Kis a tamely ramified Galois extension of number fields with Galois group isomorphic toG. The classes of rings of integers obtained in such a way for a fixed number fieldKcontaining thelth roots of unity form the subgroup of realisable classesRin the locally free class groupCl(OK[G]). On the other hand, one may also look at the Swan subgroupTofCl(OK[G]), formed by the classes of locally free OK[G]-ideals (s,Σ) wheresin OKis relatively prime toland Σ denotes the sum of the elements ofG. We show thatT(l−1)/2⊂R∩D, whereDis the kernel group ofCl(OK[G]). We also determine necessary and sufficient conditions for when a realisable class is a Swan class. Last, we show thatR∩Dis nontrivial forK=Q(ζl) whenl>3 by providing a nontrivial lower bound for the size of the Swan subgroupT.

On the Class Group Problem for Function Fields

LetGbe a finite abelian group, it is a difficult and unsolved problem to find a number fieldFwhose ideal class group is isomorphic toG. In [WAS], Corollary 3.9 and in [COR], Theorem 2, it is proved that every finite abelian group is isomorphic to a factor group of the ideal class group of some number field. Furthermore, O. Yahagi [YAH] has proved that, if l is a prime number, every finite abelian ℓ-group is isomorphic to the ℓ-Sylow subgroup of the ideal class group of some number field. In this paper, we prove similar results for the function field case.

Formal and Analytic Deformations of the Witt Algebra

For infinite dimensional Lie algebras G, the theory of formal deformations does not generally give decisive results about analytic deformations. In particular, if Gt is a family of Lie algebras depending holomorphically on t with G 0=G, and if (Gt) is formally equivalent to G[[t]], it does not necessarily follow that Gt is isomorphic toG for small ∣t∣. However, one may ask if Ĝt is isomorphic to Ĝ for small ∣t∣,Ĝ t and Ĝ being suitable completions of Gt and G respectively. In this Letter, this question is discussed when G=W, the Witt algebra.

On fixing edges in graph minors

An area of much recent research activity has been involved with tying the presence of certain minors in a matroid to specific elements of this matroid. The aim of this paper is to show that there are exactly two 3-connected simple graphsG with at least four edges and the property that ifH is a 3-connected simple graph havingG as a minor ande andf are edges ofH, thenH has a minor isomorphic toG which containse andf in its edge set. Some extensions of this result are also considered.

Chemical algebra. I: Fuzzy subgroups

Using the notion of fuzzy subset, the algebraic formulation of the constant of stereogenic pairing equilibria between skeletal analogs (previously disclosed) is connected to symmetry group theory. A distinction is introduced between geometrical (skeletal) symmetry and topographical (numerical parameters) symmetry. In order to describe “topographical symmetry”, a formal extended definition of a subgroup is proposed. Fuzzy subsets of the skeletal groupG are endowded with a structure which can be defined without referring to the geometrical representation of the abstract group isomorphic toG. The relevance of these propositions is evidenced by their “integer interpretation” meeting basic definitions of group theory, as well as by their role in expressing chemical pairing constants.

Asymptotic solution for a new class of forbiddenr-graphs

We consider the problem of finding ex (n; G), defined as the maximal number of edges anr-graph onn vertices can have that contains no subgraph isomorphic toG. We construct certainr-graphsG for which we find the coefficientτ(G) of the asymptotic expansion ex(n;G)== $$(\tau (G) + o(1))\left( {\begin{array}{*{20}c} n \\ r \\ \end{array} } \right)$$ asn→∞.

Any Banach space has an equivalent norm with trivial isometries

For any Banach spaceX there is a norm |||·||| onX, equivalent to the original one, such that (X, |||·|||) has only trivial isometries. For any groupG there is a Banach spaceX such that the group of isometries ofX is isomorphic toG × {− 1, 1}. For any countable groupG there is a norm ‖ · ‖ G onC([0, 1]) equivalent to the original one such that the group of isometries of (C([0, 1]), ‖ · ‖ G ) is isomorphic toG × {−1, + 1}.

Q-admissibility of certain nonsolvable groups

A finite groupG is calledQ-admissible if there exists a finite dimensional central division algebra overQ, containing a maximal subfield which is a Galois extension ofQ with Galois group isomorphic toG. It is proved thatS5−, one of the two nontrivial central extensions ofS5 byZ/2Z, isQ-admissible. As a consequence of that result and previous results of Sonn and Stern, every finite Sylow-metacyclic group, havingA5 as a composition factor, isQ-admissible.

On reconstructing separable reducedp-groups with a given socle

Let\(\bar B^* \) be a separable reduced (abelian)p-group which is torsion complete. We ask whether for\(G \subseteq \bar B^* \) there is\(H \subseteq _{pr} \bar B^* ,H[p] = G[p]\),H[p]=G[p],H not isomorphic toG. IfG is the sum of cyclic groups or is torsion complete, the answer is easily no. For otherG, we prove that the answer is yes assuming G.C.H. Even without G.C.H. the answer is yes if the density character ofG is equal to Minn<ω|pnG|, i.e., $$\mathop {Min}\limits_{n m} |(p^n G)[p]/(p^{n + 1} G)[p]|$$ Of course, instead of two non-isomorphic we can get many, but we do not deal much with this.

Tournaments with prescribed regular automorphism group

A tournamentX is a TRR for a groupG if (a)G acts regularly on the vertices ofX and (b) Aut(X) is isomorphic toG. We correct some previous work of Babai and Imrich by showing thatZ23 andZ33 are the only groups of odd order without TRR's. Our methods are perhaps of independent interest, since we use a probabilistic approach.

On translation planes of orderq 2 that admit a group of orderq 2 (q−1); bartolone's theorem

This article is concerned with translation planesP of orderq2 and kernelK isomorphic toG F(q). IfP admits a collineation groupG in the linear translation. complement and the order ofG K/K isq2(q−1) then it is shown thatP is either a semifield plane or is a Luneburg-Tits, Walker or Betten plane. This generalizes earlier work of Bartolone.

The norm map on Jacobians

Let π : Γ → Γ 0 \pi :\Gamma \to {\Gamma _0} be an unramified normal cover of smooth projective curves. Let π ∗ : J → J 0 {\pi _*}:J \to {J_0} be the induced map on Jacobians. Let H H be the kernel of π ∗ {\pi _*} and H 0 {H^0} the connected component of H H . We prove that H / H 0 H/{H^0} is isomorphic to G / [ G , G ] G/[G,G] where G G is the covering group of Γ / Γ 0 \Gamma /{\Gamma _0} .

Calculating the fundamental group of an orbit space

Suppose G G acts effectively as a group of homeomorphisms of the connected, locally path connected, simply connected, locally compact metric space X X . Let G ¯ \overline G denote the closure of G G in Homeo ( X ) {\text {Homeo}}(X) , and N N the smallest normal subgroup of G ¯ \overline G which contains the path component of the identity of G ¯ \overline G and all those elements of G ¯ \overline G which have fixed points. We show that π 1 ( X / G ) {\pi _1}(X/G) is isomorphic to G ¯ / N \overline G /N subject to a weak path lifting assumption for the projection X → X / G ¯ X \to X/\overline G .

On infinite tournaments with regular automorphism groups

Atournament regular representation (TRR) of an abstract groupG is a tournamentT whose automorphism group is isomorphic toG and is a regular permutation group on the vertices ofT. L. Babai and W. Imrich have shown that every finite group of odd order exceptZ3 ×Z3 admits a TRR. In the present paper we give several sufficient conditions for an infinite groupG with no element of order 2 to admit a TRR. Among these are the following: (1)G is a cyclic extension byZ of a finitely generated group; (2)G is a cyclic extension byZ2n+1 of any group admitting a TRR; (3)G is a finitely generated abelian group; (4)G is a countably generated abelian group whose torsion subgroup is finite.

On J. H. C. Whitehead's aspherical question I

A connected, finite two-dimensional CW-complex with fundamental group isomorphic toG is called a [G, 2] f -complex. LetL⊲G be a normal subgroup ofG. L has weightk if and only ifk is the smallest integer such that there exists {l1,…,lk}⊆L such thatL is the normal closure inG of {l1,…,lk}. We prove that a [G, 2] f -complexX may be embedded as a subcomplex of an aspherical complexY=X∪{e 21 ,…,e 2k } if and only ifG has a normal subgroupL of weightk such thatH=G/L is at most two-dimensional and defG=defH+k. Also, ifX is anon-aspherical [G, 2] f -subcomplex of an aspherical 2-complex, then there exists a non-trivial superperfect normal subgroupP such thatG/P has cohomological dimension ≤2. In this case, any torsion inG must be inP.

On the full automorphism group of a graph

While it is easy to characterize the graphs on which a given transitive permutation groupG acts, it is very difficult to characterize the graphsX with Aut (X)=G. We prove here that for the certain transitive permutation groups a simple necessary condition is also sufficient. As a corollary we find that, whenG is ap-group with no homomorphism ontoZ p wrZ p , almost all Cayley graphs ofG have automorphism group isomorphic toG.