Let G be a group covered by its left cosets a1G1, · · · , akGk exactly m times. is known that [G: ⋂ k i=1 Gi] � k!. When all the Gi are subnormal in G and k i=1 Gi = H, we are able to determine the least value of k in terms of m, G, H. For any i = 1, · · · , k, providing G/(Gi)G is solvable we show that k � m + f([G: Gi]) and hence [G: Gi] � 2k−m, where f(n) = ∑r s=1 αs(ps − 1) if p α1 1 · · · pαr r is the standard factorization of n. These extend some previous results on disjoint covers.

Abstract. A natural topology on the space of left orderings of an arbitrary semi-group is introduced. It is proved that this space is compact and that for free abelian groups it is homeomorphic to the Cantor set. An application of this result is a new proof of the existence of universal Gröbner bases. 1. Orderings for semi-groups Given a semi-group G (ie. a set with an associative binary operation), a linear order, <, on G is a left order if a < b implies ca < cb, for any c. Similarly, a linear order,<, is a right order if a < b implies ac < bc, for any c ∈ G. The sets of all left and right orderings of G are denoted by LO(G) and RO(G) respectively. If G is a group then there is a 1-1 correspondence between these two sets which associates with any left ordering, <l, a right ordering, <r, such that a <r b if and only if b −1 <l a −1. For more about ordering of groups see [4, 7, 8]. Let Ua,b ⊂ LO(G) denote the set of all left orderings, <, for which a < b. We can put a topology on LO(G) in one of the following two ways. Definition 1.1. LO(G) has the smallest topology for which all the sets Ua,b are open. Any open set in this topology is a union of sets of the form

Recently, operator quantum error-correcting codes have been proposed to unify and generalize decoherence free subspaces, noiseless subsystems, and quantum error-correcting codes. This note introduces a natural construction of such codes in terms of Clifford codes, an elegant generalization of stabilizer codes due to Knill. Character-theoretic methods are used to derive a simple method to construct operator quantum error-correcting codes from any classical additive code over a finite field, which obviates the need for self-orthogonal codes. Introduction. One of the main challenges in quantum information processing is the protection of the quantum information against various sources of errors. A possible remedy is given by encoding the quantum information in a subspace C of the state space H of the quantum system. If such a quantum error-correcting code C is well-chosen, then many errors can be

Abstract. This paper considers SL(n, R)-actions on Euclidean space fixing the origin. We show that all C 1-actions on R n are linearizable. We give C ∞-actions of SL(2, R) on R 3 and of SL(3, R) on R 8 which are not linearizable. We classify the C 0-actions of SL(n, R) on R n. Finally, the paper concludes with a study of the linearizability of SL(n, Z)-actions. RÉSUMÉ. Dans cet article, on considère les actions de SL(n, R) sur l’espace euclidien qui fixent l’origine. On montre que les actions C1 sur Rn sont linéarisables. On donne des actions C ∞ de SL(2, R) sur R3 et de SL(3, R) sur R8 qui ne sont pas linéarisables. On classifie les actions C0 de SL(n, R) sur Rn. L’article s’achève par une étude de la linéarisabilité des actions de SL(n, Z). 1.

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Abstract. Two ways of describing a group are considered. 1. A group is finiteautomaton presentable if its elements can be represented by strings over a finite alphabet, in such a way that the set of representing strings and the group operation can be recognized by finite automata. 2. An infinite f.g. group is quasi-finitely axiomatizable if there is a description consisting of a single first-order sentence, together with the information that the group is finitely generated. In the first part of the paper we survey examples of FA-presentable groups, but also discuss theorems restricting this class. In the second part, we give examples of quasi-finitely axiomatizable groups, consider the algebraic content of the notion, and compare it to the notion of a group which is a prime model. We also show that if a structure is bi-interpretable in parameters with the ring of integers, then it is prime and

Abstract. An iterated block cipher can be regarded as a means of producing a set of permutations of a message space. Some properties of the group generated by the round functions of such a cipher are known to be of cryptanalytic interest. It is shown here that if this group acts imprimitively on the message space then there is an exploitable weakness in the cipher. It is demonstrated that a weakness of this type can be used to construct a trapdoor that may be difficult to detect. An example of a DES-like cipher, resistant to both linear and differential cryptanalysis that generates an imprimitive group and is easily broken, is given. Some implications for block cipher design are noted. 1

Abstract. Let N be a closed, oriented 3–manifold. A folklore conjecture states that S 1 ×N admits a symplectic structure only if N admits a fibration over the circle. The purpose of this paper is to provide evidence to this conjecture studying suitable twisted Alexander polynomials of N, and showing that their behavior is the same as of those of fibered 3–manifolds. In particular, we will obtain new obstructions to the existence of symplectic structures and to the existence of symplectic forms representing certain cohomology classes of S 1 ×N. As an application of these results we will show that S 1 ×N(P) does not admit a symplectic structure, where N(P) is the 0–surgery along the pretzel knot P = (5, −3, 5), answering a question of Peter Kronheimer. 1.

A nonidentity element of a permutation group is said to be semiregular

Abstract. We prove that the solvable radical of a finite group G coincides with the set of elements y having the following property: for any x ∈ G the subgroup of G generated by x and y is solvable. We present analogues of this result for finite dimensional Lie algebras and some classes of infinite groups. To Charles Leedham-Green on his 65th birthday 1.