In this paper we are concerned with the conjecture that, for any set of generators S of the symmetric group Sym(n), the word length in terms of S of every permutation is bounded above by a polynomial of n. We prove this conjecture for sets of generators containing a permutation fixing at least 37% of the points.

A nite set of generators for the mapping class group of a punctured nonorientable surface is given. When the surface has at least two punctures, up to conjugates there are four generators; a Dehn twist about a two-sided nonseparating simple closed curve, a crosscap slide, a boundary slide and an elementary braid.

In this paper we are concerned with the conjecture that, for any set of generators S of the symmetric group Sym(n), the word length in terms of S of every permutation is bounded above by a polynomial of n. We prove this conjecture for sets of generators containing a permutation fixing at least 37% of the points.

We give necessary and sufficient conditions for the first-order theory of a finitely presented abelian lattice-ordered group to be decidable. We also show that if the number of generators is at most 3, then elementary equivalence implies isomorphism. We deduce from our methods that the theory of the free MV -algebra on at least 2 generators is undecidable.

We present a detailed study of the representations of the algebra of functions on the quantum group GLq(n). A q-analouge of the root system is constructed for this algebra which is then used to determine explicit matrix representations of the generators of this algebra. At the end a q-boson realization of the generators of GLq(n) is given.

In the paper an upper bound is established for certain exponential sums, analogous to Gaussian sums, defined on the points of an elliptic curve over a prime finite field. The bound is applied to prove the existence of group generators for the set of points on an elliptic curve over Fq among certain sets of bounded size. We apply this estimate to obtain a deterministic O(q) algorithm for finding...

Let Fn be the free group on n generators, and let PΣn be the group of automorphisms of Fn that send each generator to a conjugate of itself. The kernel Kn of the homomorphism PΣn → PΣn−1, induced by mapping one of the free group generators to the identity, is finitely generated. We show that Kn has cohomological dimension n − 1, and that Hi(Kn;Z) is not finitely generated for 2 ≤ i ≤ n − 1. It ...

Let Fn denote the free group generated by n letters. The purpose of this article is to show that Hol(F2), the holomorph of the free group on two generators, is linear. Consequently, any split group extension G = F2 ⋊H for which H is linear has the property that G is linear. This result gives a large linear subgroup of Aut(F3). A second application is that the mapping class group for genus one s...

We give a modification of I. Klep and M. Schweighofer algebraic reformulation of Connes’ embedding problem by considering ∗-algebra of the countably generated free group. This allows to consider only quadratic polynomials in unitary generators instead of arbitrary polynomials in self-adjoint generators.