齐次可分辨的完全可约的线性变换代数的张量积

Jacobson在〔1,ch6〕中,给出了如下定义:M为一加群,A是{M,+}的自同态环,则M可以看成是忠实右A-模。如M是完全可约的A-模,则称A为完全可约的自同态环;如A还是齐次的,即M的所有不可约A-子模均同构,则称A是齐次完全可约的自同态环;如完全可约自同态环A在M所定义的有限拓扑中是闭的,则称A是可分辨的完全可约自同态环。     

On finite groups with a reducible permutability-graph

We characterize finite groups in which the permutability-graph has more than one connected component.Research partially supported by G.N.S.A.G.A. of C.N.R. and M.U.R.S.T. of Italy.     

Normal bundle to subvarieties in quadratics II

Summary In this paper we prove that the restriction of the tangent bundle of a nonsingular quadrix Q to a subvariety X is ample if and only if X does not contain a straight line. This implies that the normal bundle of a locally complete intersection, reduced and irreducible curve C is ample if and only if C is not a straight line. The result gives information also for higher dimensional subvarieties of Q.The author is member of G.N.S.A.G.A. of C.N.R.     

Two applications of maschke's theorem

Let R ? G denote a crossed product of the finite group G over the ring R and let V be an R ? G-module. Maschke's theorem states that if 1/∣G∣ ε R and if V is completely reducible as an R-module, then V is also completely reducible as an R ? G -module. In this paper, we obtain two applications of this theorem, both under the assumption that R is semiprime with no ∣G∣ -torsion. The first concerns group actions and here we show that if G acts on R and if I is an essential right ideal of the fixed ring R^G , then IR is essential in Rs. This result, in turn, simplifies a number of proofs already in the literature. The second application here is a short proof of a theorem of Fisher and Montgomery which asserts that the crossed product R ? G is semiprime.     

关于Glauberman-Isaacs特征标对映的注记

假设群A经自同构互素地作用在G上.设χ是G的一个A-不变不可约特征标,π(G,A)表示Glauberman-Isaacs特征标对映.对于B≤A,T.R.Wolf曾猜想χπ(G,A)是χπ(G,B)a的一个不可约成份,此处C=CG(A).设G=N(X)H且(|N|,|H|)=1,假定H是A-不变的且N是一个Sylow塔群,N的Sylow-子群是交换的.在本文中,我们证明了如果这个猜想对所有H的A-不变子群成立,则猜想对G也成立.     

Vertex operator algebras,generalized doubles and dual pairs

Let V be a simple vertex operator algebra and G a finite automorphism group. Then there is a natural right G-action on the set of all inequivalent irreducible V-modules. Let be a finite set of inequivalent irreducible V-modules which is closed under the action of G. There is a finite dimensional semisimple associative algebra for a suitable 2-cocycle naturally determined by the G-action on such that and the vertex operator algebra form a dual pair on the sum of V-modules in in the sense of Howe. In particular, every irreducible V-module is completely reducible -module. Received: 10 September, 2001 / Published online: 29 April 2002 RID="*" ID="*" Supported by NSF grants and a research grant from the Committee on Research, UC Santa Cruz. RID="**" ID="**" Supported by DPST grant from government of Thailand.     

2-Length and rational characters of odd degree

If G is a finite solvable group of 2-length l, we prove that the number of odd degree rational-valued irreducible characters of G is at least 2 ^l , improving a result of G. Navarro and P. H. Tiep. This bound is best possible, and also provides a new global/local relationship.     

Isomorfismi tra le strutture subnormali dei gruppi

Summary In this paper order-isomorphisms between subnormal structures of subsoluble groups are considered, and the images of generalized nilpotent groups in such isomorphisms are studied. A result of Pazderski about the Fitting subgroup of finite soluble groups is also extended to the upper Baer series of subsoluble groups, and an extension to infinite groups of a theorem of Heineken about isomorphisms between lattices of subnormal subgroups of finite groups is given.

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Lavoro eseguito nell'ambito del G.N.S.A.G.A. del C.N.R.     

Transitive groups with irreducible representations of bounded degree

A well-known theorem of Jordan states that there exists a function J(d) of a positive integer d for which the following holds: if G is a finite group having a faithful linear representation over ℂ of degree d, then G has a normal Abelian subgroup A with [G:A]≤J(d). We show that if G is a transitive permutation group and d is the maximal degree of irreducible representations of G entering its permutation representation, then there exists a normal solvable subgroup A of G such that [G:A]≤J(d) ^log ₂ ^d. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 108–119. Translated by S. A. Evdokimov.     

On the Lüroth semigroup of curves lying on a smooth quadric

For a smooth irreducible complete algebraic curveC the “gaps” are the integersn such that every linear series of degreen has at least a base point. The Lüroth semigroup S_C of a curveC is the subsemigroup ofN whose elements are not gaps. In this paper we deal with irreducible smooth curves of type (a, b) on a smooth quadricQ. The main result is an algorithm by which we can say if some integer λ∈N is a gap or is in S_C. In the general case there are integers λ which are undecidable. For curves such as complete intersection, arithmetically Cohen-Macaulay or Buchsbaum, we are able to describe explicitly “intervals” of gaps and “intervals” of integers which belong to S_C. For particular cases we can completely determine S_C, by giving just the type of the curve (in particular the degree and the genus). Work done with financial support of M.U.R.S.T. while the authors were members of G.N.S.A.G.A. of C.N.R.     

Forbidden minors for wye‐delta‐wye reducibility

A graph is Y Δ Y reducible if it can be reduced to a single vertex by a sequence of series‐parallel reductions and Y Δ Y transformations. The class of Y Δ Y reducible graphs is minor closed. We found a large number of minor minimal Y Δ Y irreducible graphs: a family of 57578 31‐edge graphs and another 40‐edge graph. It is still an open problem to characterize Y Δ Y reducible graphs in terms of a finite set of forbidden minors. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 317–321, 2004     

The transvection free groups with polynomial rings of invariants

Let G be a finite linear group containing no transvections. This paper proves that the ring of invariants of G is polynomial if and only if the pointwise stabilizer in G of any subspace is generated by pseudoreflections. Kemper and Malle used the classification of finite irreducible groups generated by pseudoreflections to prove the irreducible case in arbitrary characteristic. We extend their result to the reducible case.     

Strutture di André edS-spazi con traslazioni: II

Summary We investigate finite André-structures and Sperner-spaces with the property that the stabilizer of a line in the traslation group is never identical. For this purpose we make use of a suitable representation of these structures by means of a set of partitions of a finite group. Results of various types are obtained, mostly in connection with collineations and constructions of new classes.

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Lavoro eseguito nell'ambito delle attività del G.N.S.A.G.A. del C.N.R.     

关于有限群的不可约特征标个数的一个结果

本文研究了一般的非交换有限群G的阶与不可约特征标个数的比值与群G结构之间的关系.通过群G阶的最小素因子和G的换位子群的阶的最小素因子,得出了这个比值的下界,并确定了达到下界的一个充分必要条件.     

An algorithm and self-reproducing systems

Let G be a group and let N be a normal subgroup of G. We set cd(G|N) to be the degrees of the irreducible characters of G whose kernels do not contain N. We associate a graph with this set. The vertices of this graph are the primes dividing degrees in cd(G|N), and there is an edge between p and q if pq divides some degree in cd(G|N). In this paper, we study this graph when it is disconnected, and we study its diameter when it is connected.     

On large orientation-reversing finite group-actions on 3-manifolds and equivariant Heegaard decompositions

We consider finite group-actions on closed, orientable and nonorientable 3-manifolds; such a finite group-action leaves invariant the two handlebodies of a Heegaard splitting of M of some genus g. The maximal possible order of a finite group-action of an orientable or nonorientable handlebody of genus $$g>1$$ is $$24(g-1)$$, and in the present paper we characterize the 3-manifolds M and groups G for which the maximal possible order $$|G| = 24(g-1)$$ is obtained, for some G-invariant Heegaard splitting of genus $$g>1$$. If M is reducible then it is obtained by doubling an action of maximal possible order $$24(g-1)$$ on a handlebody of genus g. If M is irreducible then it is a spherical, Euclidean or hyperbolic manifold obtained as a quotient of one of the three geometries by a normal subgroup of finite index of a Coxeter group associated to a Coxeter tetrahedron, or of a twisted version of such a Coxeter group.     

Positions of characters in finite groups and the Taketa inequality

We define the position of an irreducible complex character of a finite group as an alternative to the degree. We then use this to define three classes of groups: position reducible (PR)-groups, inductively position reducible (IPR)-groups and weak IPR-groups. We show that IPR-groups and weak IPR-groups are solvable and satisfy the Taketa inequality (ie, that the derived length of the group is at most the number of degrees of irreducible complex characters of the group), and we show that any M-group is a weak IPR-group. We also show that even though PR-groups need not be solvable, they cannot be perfect.     

Su una condizione di fattorialità debole e l’annullamento del gruppo di Picard

Summary We give some characterizations of noetherian domains A such that ? every irreducible element generates a primary ideal ?. This condition, called (α)-property, is equivalent to the unique factorization if A is normal or a polynomial ring A=B[T]. If A is a1-dimensional k-algebra, the property (α) is equivalent to the vanishing of some Picard groups asPicA,Pic (A[T, T⁻¹]),Pic (A|T|_s), where S={Tⁿ, n εZ}. We give not trivial examples of (α)-rings which aren’t factorial.

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Entrata in Redazione il 6 febbraio 1976.

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Lavoro eseguito nell’ambito della sezione n. 3 del G.N.S.A.G.A. del C.N.R.