We present previously unpublished elementary proofs by Dekker and Ottens (1991) and Boyce (private communication) of a special case of the Dinitz conjecture. We prove a special case of a related basis conjecture by Rota, and give a reformulation of Rota's conjecture using the Nullstellensatz. Finally we give an asymptotic result on a related Latin square conjecture.

The best known asymptotic bit complexity bound for factoring univariate polynomials over finite fields grows with $$d^{1.5 + o (1)}$$ input of degree d, and the square size ground field. It relies on a variant Cantor–Zassenhaus algorithm which exploits fast modular composition. Using techniques by Kaltofen Shoup, we prove refinement this when field has large extension its prime We also present ...

Suppose R is a complete local Dedekind domain with quotient field F, and let f(x) be a monic polynomial in R[x] having non-zero discriminant. We present here a new algorithm to construct the maximal order of the algebra Af = F[x]/f(x)F[x]. The new algorithm incorporates ideas of Zassenhaus (1975, 1980) concerning P-adic stability and the algebraic decomposition of A s . We show that it is alway...

2. G. Higman, On finite groups of exponent five, Proc. Cambridge Philos. Soc. 52 (1956), 381-390. 3. A. I. Kostrikin, On Burnside's problem, Dokl. Akad. Nauk SSSR 119 (1958), 1081-1084. (Russian) 4. M. Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. École Norm. Sup. (3) 71 (1954), 101-190. 5. H. Zassenhaus, Ein Verfahren, jeder endlichen p-Gruppe eine Lie-Ring mit der Charakteris...

We determine the number ω(G) of orbits on the (finite) group G under the action of Aut(G) for G ∈ {PSL(2, q),SL(2, q),PSL(3, 3),Sz(2)}, covering all of the minimal simple groups as well as all of the simple Zassenhaus groups. This leads to recursive formulae on the one hand, and to the equation ω(Sz(q)) = ω(PSL(2, q)) + 2 on the other. MSC 20E32, 20F28, 20G40, 20-04

2. (Jordan) A transitive permutation group of degree n > 1 contains a derangement. In fact (Cameron and Cohen) the proportion of derangements in a transitive group G is at least 1/n. Equality holds if and only if G is sharply 2transitive, and hence is the affine group {x 7→ ax + b : a, b ∈ F, a 6= 0} over a nearfield F. The finite nearfields were determined by Zassenhaus. They all have prime po...

The goal of this article is to formalize the Jordan-Hölder theorem in the context of group with operators as in the book [5]. Accordingly, the article introduces the structure of group with operators and reformulates some theorems on a group already present in the Mizar Mathematical Library. Next, the article formalizes the Zassenhaus butterfly lemma and the Schreier refinement theorem, and def...

Recently, a number of new classes of infinite-dimensional simple Lie algebras over a Field of characteristic 0 were discovered by several authors (see the references at the end of this paper). Among those algebras, are the generalized Witt algebras. The higher rank Virasoro algebras was introduced by Patera and Zassenhaus [PZ], which are 1-dimensional universal central extensions of some genera...

The algorithm for factoring polynomials over the integers by Wang and Rothschild is generalized to an algorithm for the irreducible factorization of multivariate polynomials over any given algebraic number field. The extended method makes use of recent ideas in factoring univariate polynomials over large finite fields due to Berlekamp and Zassenhaus. The procedure described has been implemented...