Fusions of Character Tables and Schur Rings of Abelian Groups

If the character table of a finite group H satisfies certain conditions, then the classes and characters of H can fuse to give the character table of a group G of the same order. We investigate the case where H is an abelian group. The theory is developed in terms of the S-rings of Schur and Wielandt. We discuss certain classes of p-groups which fuse from abelian groups and give examples of such groups which do not. We also show that a large class of simple groups do not fuse from abelian groups. The methods to show fusion include the use of extensions which are Camina pairs, but other techniques on S-rings are also developed.     

Fusions of character tables III: Fusions of cyclic groups and a generalisation of a condition of Camina

In this paper we characterise finite groups whose character tables fuse from the character table of a cyclic group. We also show a connection with a generalisation of the Camina pair condition introduced by Camina in [Ca].     

Transitive permutation groups in which all derangements are involutions

Let G be a transitive permutation group in which all derangements are involutions. We prove that G is either an elementary abelian 2-group or is a Frobenius group having an elementary abelian 2-group as kernel. We also consider the analogous problem for abstract groups, and we classify groups G with a proper subgroup H such that every element of G not conjugate to an element of H is an involution.     

The linking pairings of orientable Seifert manifolds

We compute the p-primary components of the linking pairings of orientable 3-manifolds admitting a fixed-point free S¹-action. Any linking pairing on a finite abelian group of odd order is realized by such a manifold. We find necessary and sufficient conditions for a pairing on an abelian 2-group to be the 2-primary component of such a linking pairing, and give simple examples which are not realizable by any Seifert fibred 3-manifold.     

Minimal Number of Generators and Minimum Order of a Non-Abelian Group Whose Elements Commute with Their Endomorphic Images

A group in which every element commutes with its endomorphic images is called an “E-group″. If p is a prime number, a p-group G which is an E-group is called a “pE-group″. Every abelian group is obviously an E-group. We prove that every 2-generator E-group is abelian and that all 3-generator E-groups are nilpotent of class at most 2. It is also proved that every infinite 3-generator E-group is abelian. We conjecture that every finite 3-generator E-group should be abelian. Moreover, we show that the minimum order of a non-abelian pE-group is p ⁸ for any odd prime number p and this order is 2⁷ for p = 2. Some of these results are proved for a class wider than the class of E-groups.     

Generalizing alternative rings

We give an example of two groups G and H, generated by 2 and 3 elements, which satisfy G × Z ? H × Z.The groups G, H are semi-direct products of a torsion-free abelian group of rank 3 by an infinite cyclic group.     

Conjugacy Classes,Class Sums and Character Tables for Hopf Algebras

We extend the notion of conjugacy classes and class sums from finite groups to semisimple Hopf algebras and show that the conjugacy classes are obtained from the factorization of H as irreducible left D(H)-modules. For quasitriangular semisimple Hopf algebras H, we prove that the product of two class sums is an integral combination of the class sums up to d ^?2 where d = dim H. We show also that in this case the character table is obtained from the S-matrix associated to D(H). Finally, we calculate explicitly the generalized character table of D(kS ₃), which is not a character table for any group. It moreover provides an example of a product of two class sums which is not an integral combination of class sums.     

Completion of Linearly Ordered Groups

We give examples of linearly ordered groups that are not embeddable in divisible orderable. In the first example, the group does not embed in any divisible group with strictly isolated unity. In the second example, the group in question is an O*-group, and in the third, it is a group with a central system of convex subgroups. To my teacher A. I. Kokorin Supported by RFBR grant Nos. 96-01-00358, 99-01-00335, and 03-01-00320. __________ Translated from Algebra i Logika, Vol. 44, No. 6, pp. 664–681, November–December, 2005.     

Topologies arising from metrics valued in abelian <Emphasis Type="Italic">ℓ</Emphasis>-groups

This paper considers metrics valued in abelian ℓ-groups and their induced topologies. In addition to a metric into an ℓ-group, one needs a filter in the positive cone to determine which balls are neighborhoods of their center. As a key special case, we discuss a topology on a lattice ordered abelian group from the metric d _G and the positive filter consisting of the weak units of G; in the case of \mathbb Rⁿ{\mathbb R^{n}} , this is the Euclidean topology. We also show that there are many Nachbin convex topologies on an ℓ-group which are not induced by any positive filter of the ℓ-group.     

Galois representations,Mumford-Tate groups and good reduction of abelian varieties

Let K be a number field and A an abelian variety over K. We are interested in the following conjecture of Morita: if the Mumford-Tate group of A does not contain unipotent -rational points then A has potentially good reduction at any discrete place of K. The Mumford-Tate group is an object of analytical nature whereas having good reduction is an arithmetical notion, linked to the ramification of Galois representations. This conjecture has been proved by Morita for particular abelian varieties with many endomorphisms (called of PEL type). Noot obtained results for abelian varieties without nontrivial endomorphisms (Mumfords example, not of PEL type). We give new results for abelian varieties not of PEL type.An erratum to this article can be found at     

Some remarks on almost l-groups

The divisibility group of every Bézout domain is an abelian l-group. Conversely, Jaffard, Kaplansky, and Ohm proved that each abelian l-group can be obtained in this way, which generalizes Krull’s theorem for abelian linearly ordered groups. Dumitrescu, Lequain, Mott, and Zafrullah [3] proved that an integral domain is almost GCD if and only if its divisibility group is an almost l-group. Then they asked whether the Krull-Jaffard-Kaplansky-Ohm theorem on l-groups can be extended to the framework of almost l-groups, and asked under what conditions an almost l-group is lattice-ordered [3, Questions 1 and 2]. This note answers the two questions. Received: 29 April 2008     

How often can a finite group be realized as a Galois group over a field?

Let G be any finite group and any class of fields. By we denote the minimal number of realizations of G as a Galois group over some field from the class . For G abelian and the class of algebraic extensions of ℚ we give an explicit formula for . Similarly we treat the case of an abelian p-group G and the class which is conjectured to be the class of all fields of characteristic ≠p for which the Galois group of the maximal p-extension is finitely generated. For non-abelian groups G we offer a variety of sporadic results. Received: 27 October 1998 / Revised version: 3 February 1999     

Construction of Ternary H v -groups and Ternary P-hyperoperations

In this paper, we study the concept of ternary H _v -groups and some their properties. We give some examples of ternary H _v -groups. Also, we consider the fundamental relation β* on a ternary H _v -group and prove that β* is a compatible relation on a ternary H _v -group. In addition, we define the P-hyperoperation, and then, we construct a new ternary H _v -group.     

Automorphism Groups of Finite Abelian p-groups and Transvections

We define a particular type of automorphisms called transvections on a finite finite abelian p-group H_p. It is proved that the subgroup E of the automorphism group Aut(H_p) of H_p generated by those transvections is normal in it, and that Aut(H_p) can be written as the product of E and some abelian subgroup K. The center of Aut(H_p) is also determined.     

Transfer and Tate’s theorem

Let H be a subgroup of a finite group G, and assume that p is a prime that does not divide |G : H|. In favorable circumstances, one can use transfer theory to deduce that the largest abelian p-groups that occur as factor groups of G and of H are isomorphic. When this happens, Tate’s theorem guarantees that the largest not-necessarily-abelian p-groups that occur as factor groups of G and H are isomorphic. Known proofs of Tate’s theorem involve cohomology or character theory, but in this paper, a new elementary proof is given. It is also shown that the largest abelian p-factor group of G is always isomorphic to a direct factor of the largest abelian p-factor group of H. Received: 17 June 2008     

Some cases of preservation of the Pontryagin dual by taking dense subgroups

We study the compact-open topology on the character group of dense subgroups of topological abelian groups. Permanence properties concerning open subgroups, quotients and products are considered. We also present some representative examples. We prove that every compact abelian group G with w(G)?c has a dense pseudocompact group which does not determine G; this provides (under CH) a negative answer to a question posed by S. Hernández, S. Macario and the third listed author two years ago.     

Groups with trivial virtual automorphism group

We answer a question of A. Lubotzky and A. Mann by constructing examples of infinite groupsG such that every isomorphismα:H →K between subgroupsH andK having finite index inG coincides with the identity on some subgroup of finite index. The structure of such a group is very restricted;G must be virtually a 2-group with finite central derived subgroup andG/G′ elementary abelian. This work was begun while the second author was a visitor at the University of Padova. He wishes to thank the Mathematics Department for its hospitality and the C.N.R. for its financial support.