图的字典序积和自同态幺半群

Ｆ．Ｈａｒａｒｙ￣［１］和Ｇ．Ｓａｂｉｄｕｓｓｉ￣［２］考虑过图Ｘ和ｙ的字典序积Ｘ［Ｙ］的自同构群ＡｕｔＸ［Ｙ］与它们各自的自同构群的圈积ＡｕｔＸ［ＡｕｔＹ］的关系，并给出了两者相等的一种刻划．在本文，我们考虑更广意义上的问题，即Ｘ［Ｙ］的自同态幺半群ＥｎｄＸ［Ｙ］与各自的自同态幺半群的圈积ＥｎｄＸ［ＥｎｄＹ］的关系，也给出了两者相等的一种刻划，同时得到了下面结果：如果Ｘ和Ｙ都是不含Ｋ＿３导出子图的连通图，且其中之一图有奇数围长，那么ＥｎｄＸ［Ｙ］＝ＥｎｄＸ［ＥｎｄＹ］．     

Commutator Calculus for Wreath Product Groups

We introduce a class of function spaces consisting of integer-valued functions of several integers which coordinatize the elements of certain subgroups of some finite regular wreath product groups. On each function space, we define operators which correspond to forming certain commutators relevent to computing the upper central series. We define an automorphism of the function space which enables us to define a class of subgroups that is useful for describing the upper central series of certain finite regular wreath product p-groups. Our results describe fundamental, interesting, and useful relationships between the automorphism and the operators. We describe some applications.     

Automorphism group and dimension of ordered sets

It is shown that a finite groupG is isomorphic to the automorphism group of a two-dimensional ordered set if and only if it is a generalized wreath product of symmetric groups over an ordered index set that is a dual tree. Furthermore, every finite abelian group is isomorphic to the full automorphism group of a three-dimensional ordered set. Also every finite group is isomorphic to the automorphism group of an ordered set that does not contain an induced crown with more than four elements.     

Semicomplete Permutational Wreath Products

A group is called semicomplete if every automorphism which induces the identity on the factor commutator group is inner. In this paper, we study the connection of the semicompleteness of the permutational wreath product W of two groups with the semicompleteness of these groups. We give necessary conditions under which the group W is semicomplete.2000 Mathematics Subject Classification: 20E22, 20E36     

Coleman Automorphisms of Permutational Wreath Products

Let N be a finite nontrivial nilpotent group and H a finite centerless permutation group on a finite set Ω (i.e., H acts faithfully on Ω). Let G = N?H = N^|Ω| ? H be the corresponding permutational wreath product of N by H. It is shown that every Coleman automorphism of G is an inner automorphism. This generalizes a well-known result due to Petit Lobão and Sehgal stating that the normalizer property holds for complete monomial groups with nilpotent base groups.     

The structure of ?*-inverse semigroups

The concepts of ℒ*-inverse semigroups and left wreath products of semigroups are introduced. It is shown that the ℒ*-inverse semigroup can be described as the left wreath product of a type A semigroup Γ and a left regular band B together with a mapping which maps the semigroup Γ into the endomorphism semigroup End(B). This result generalizes the structure theorem of Yamada for the left inverse semigroups in the class of regular semigroups. We shall also provide a constructed example for the ℒ*-inverse semigroups by using the left wreath products.     

Double centralizing theorem for wreath products of the alternating group

The double centralizing theorem between the action of the symmetric group S_n and the action of the general linear group on the tensor space T_n(W) was obtained by Schur. Here we obtain a double centralizing theorem when S_n is replaced by the wreath product of a finite group G and the alternating group A_n.     

A proof of a conjecture of Sabidussi on graphs idempotent under the lexicographic product

In 1960, Sabidussi conjectured that if a graph G is isomorphic to the lexicographic product G[G], then the wreath product of by itself is a proper subgroup of . A positive answer is provided by constructing an automorphism Ψ of G[G] which satisfies: for every vertex x of G, there is an infinite subset I(x) of V(G) such that Ψ({x}×V(G))=I(x)×V(G).     

Wreath products of the groups of monotone permutations

We introduce the concept of wreath product of the m-groups of permutations and prove that an m-transitive group of permutations with an m-congruence is embeddable into the wreath product of the suitable m-transitive m-groups of permutations. This implies that an arbitrary m-transitive group in the product of two varieties of m-groups embeds into the wreath product of the suitable m-transitive groups of these varieties.     

Factoring cartesian-product graphs

In a fundamental paper, G. Sabidussi [“Graph Multiplication,” Mathematische Zeitschrift, Vol. 72 (1960), pp. 446–457] used a tower of equivalence relations on the edge set E(G) of a connected graph G to decompose G into a Cartesian product of prime graphs. Later, a method by R.L. Graham and P.M. Winkler [“On Isometric Embeddings of Graphs,” Transactions of the American Mathematics Society, Vol. 288 (1985), pp. 527–533] of embedding a connected graph isometrically into Cartesian products opened another approach to this problem. In both approaches an equivalence relation σ that determines the prime factorization is constructed. The methods differ by the starting relations used. We show that σ can be obtained as the convex hull of the starting relation used by Sabidussi. Our result also holds for the relation determining the prime decomposition of infinite connected graphs with respect to the weak Cartesian product. Moreover, we show that this relation is the transitive closure of the union of the starting relations of Sabidussi and Winkler [“Factoring a Graph in Polynomial Time,” European Journal of Combinatorics, Vol. 8 (1987), pp. 209–212], thereby generalizing the result of T. Feder [“Product Graph Representations,” Journal of Graph Theory, Vol 16 (1993), pp. 467–488] from finite to infinite graphs.     

An Embedding Theorem for Automorphism Groups of Cartan Geometries

We study the automorphism group of a Cartan geometry, and prove an embedding theorem analogous to a result of Zimmer for automorphism groups of G-structures. Our embedding theorem leads to general upper bounds on the real rank or nilpotence degree of a Lie subgroup of the automorphism group. We prove that if the maximal real rank is attained in the automorphism group of a geometry of parabolic type, then the geometry is flat and complete.     

Centralizers in symmetric inverse semigroups: Structure and order

The representation [5] of the centralizerC(x) of a permutationx in (a symmetric inverse semigroup)C _n involves direct products of wreath products. Indeed, this semigroup case extends its group theory counterpart. Here, the last case (forx nilpotent) is addressed: A quotient of a wreath product is introduced and used to obtain a representation of the correspondingC(x). It follows that, for anyx∈C _n ,C(x) can be imbedded in a direct product of wreath products with a quotient of a wreath product. A formula for calculating the order ofC(x) is given. The independent parameters in the formula are precisely those that define the path structure ofx∈C _n . Part of this research was supported by a Mary Washington College faculty development grant.     

Free Wreath Product by the Quantum Permutation Group

Let A be a compact quantum group, let nN ^* and let A _aut(X _n ) be the quantum permutation group on n letters. A free wreath product construction A*_w A _aut(X _n ) is introduced. This construction provides new examples of quantum groups, and is useful to describe the quantum automorphism group of the n-times disjoint union of a finite connected graph.     

Reflexible maps and hypermaps with face–stabilisers pointwise stabilising half the faces

This paper deals mainly with reflexible hypermaps in which the stabiliser of a hyperface fixes exactly half the hyperfaces - these reflexible hypermaps are called here 2-dichromatic. The number of hyperfaces of any 2-dichromatic hypermap must be necessarily even and greater than or equal to 4. We prove that if then is necessarily orientable and of type , for some positive integers , and , and show that the automorphism group of a 2-dichromatic hypermap is a wreath product. We also construct an infinite family of orientable 2-dichromatic hypermaps of type with 2n hyperfaces (n even). If is a 2-dichromatic map then . In 1959 Sherk [19] described an infinite family of orientable maps, he denoted by , where , and are positive integers satisfying certain conditions. We find in the dual family a subfamily of infinitely many 2-dichromatic maps. Received 23 August 1999; revised 27 March 2000.     

On automorphism groups of quasiprimitive 2-arc transitive graphs

We characterize the automorphism groups of quasiprimitive 2-arc-transitive graphs of twisted wreath product type. This is a partial solution for a problem of Praeger regarding quasiprimitive 2-arc transitive graphs. The solution stimulates several further research problems regarding automorphism groups of edge-transitive Cayley graphs and digraphs. This work forms part of an ARC grant project and is supported by a QEII Fellowship.     

Finite state wreath powers of transformation semigroups

The finite state wreath power of a transformation semigroup is introduced. It is proved that the finite state wreath power of nontrivial semigroup is not finitely generated and in some cases even does not contain irreducible generating systems. The free product of two monogenic semigroups of index 1 and period m is constructed in the finite state wreath power of corresponding monogenic monoid.     

Quasivarieties of groups closed under wreath products and wreathZ-products

LetL _q(M) be a lattice of quasivarieties contained in a quasivarietyM. The quasivariety is closed under direct wreath Z-products if together with a group G, it contains its wreath product G ≀ Z with an infinite cyclic group Z. We prove the following: (a) ifM is closed under direct wreath Z-products then every quasivariety, which is a coatom inL _q(M), is likewise closed under these; (b) ifM is closed under direct wreath products thenL _q(M) has at most one coatom. An example of a quasivariety is furnished which is closed under direct wreath Z-products and whose subquasivariety lattice contains exactly one coatom. Also, it turns out that the set of quasivarieties closed under direct wreath Z-products form a complete sublatttice of the lattice of quasivarieties of groups. Supported by RFFR grant No. 96-01-00088, and by the RF Committee of Higher Education. Translated fromAlgebra is Logika, Vol. 38, No. 3, pp. 257–268, May–June, 1999.     

On unretractive graphs

The present paper proves necessary and sufficient conditions for both lexicographic products and arbitrary graphs to be unretractive. The paper also proves that the automorphism group of a lexicographic product of graphs is isomorphic to a wreath product of a monoid with a small category.     

The Number of Cyclic Configurations of Type (v3) and the Isomorphism Problem

A configuration of points and lines is cyclic if it has an automorphism that permutes its points in a full cycle. A closed formula is derived for the number of nonisomorphic connected cyclic configurations of type (v₃), i.e. which have v points and lines, and each point/line is incident with exactly three lines/points. In addition, a Bays–Lambossy type theorem is proved for cyclic configurations if the number of points is a product of two primes or a prime power.     

Isomorphisms of integral alternative loop rings

This paper first settles the “isomorphism problem” for alternative loop rings; namely, it is shown that a Moufang loop whose integral loop ring is alternative is determined up to isomorphism by that loop ring. Secondly, it is shown that every normalized automorphism of an alternative loop ringZ L is the product of an inner automorphism ofQ L and an authomorphism ofL.