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Throughout this note K will denote a field of characteristic p > 0. Let I be the set {1,2, … , m}, and a finite additive group of functions on I with values in K. We assume that is total in the sense that, for any λ1, … , λm in K Σi=imλiσ(i) = 0 for all a in G implies all σi = 0. It is clear that is an elementary p-group. Let pn be the order of . A generalized Witt algebra is defined as an algebra over K with basis elements {e(σ, i)∣ σ ∈ , i ∈ I} and the multiplication table is a simple Lie algebra except when p = 2, m = 1.

Introduction. We study a family of Lie algebras of characteristic p which are defined as subalgebras of the derivation algebra of the group algebra of an elementary p-group. In particular we show that simple Lie algebras of dimensions m (pn -1), mpn, pn_ 2, where m and n are arbitrary integers such that 1 2 only for the dimensions pn and pn 2, are associated with this family. The algebras studied by M. S. Frank [2] are included in our family, but those of dimension m(pn -1) in general appear to be new. Since this paper was written, the paper of A. A. Albert and M. S. Frank

Let λ(ij), i,j = 1, 2, … , m, be m2 elements in a field K of characteristic zero such that λ(ij)λ(ji) = 1 for all i and j, and X1, x2, … , xm non-commutative associative indeterminates over K. Define the elements [xi1Xi2 … xin] inductively by [xi] = xi andAny linear combination of the elementswith coefficients in K will be called a generalized Lie elememt. Generalized Lie elements reduce to ordinary Lie elements if λ(ij) = 1 for all i and j.The purpose of this paper is to generalize to the generalized Lie elements the following: a theorem of Friedrichs, a theorem of Dynkin-Specht-Wever (2), and the Witt formula on the dimension of the space spanned by homogeneous Lie elements of a fixed degree. The set of all generalized Lie elements will be made into an algebra which generalizes the ordinary free Lie algebra. This algebra turns out to be free in a certain sense. We shall also generalize the algebra associated with shuffles in (2).

Denote by Fα,β the multiplicative group generated by the two matrices where α and β are complex numbers. Sanov (3) proved that F α,β is free, and Brenner (1) showed that F m,m is free if m ⩾ 2.

Introduction. Let V be an additively written elementary p-group of order pn, where n > 1, and let 4(u, v), VIt(u, v) be two alternate biadditive functions defined on V and with values in a field K of characteristic p. Let k be a fixed nonzero element in V, and assume that the following two conditions are satisfied: (0.1) If a, jB are nonzero elements in K then oa4(u, v) +/p3(u, v) =0 for all v implies u=0; (0.2) sb(k, v)=0 for all v. If 0 is identically zero, then the conditions (0.1)-(0.2) are equivalent to saying that i& is nondegenerate. Define an algebra L over K with basis

A complex number λ will be said to be free if the multiplicative group Fλ generated by the two matricesis a free group, and non-free, otherwise. Very little is known about the distribution of free and non-free numbers [1]. It is, for instance, unknown whether the domaincontains an open set which consists of only free points.

This paper proves the following theorem. LetG be a group generated byA andB. LetW be a word inA andB which in cyclically reduced form contains bothA andB. IfG has the presentationG=, u ≳=2, then there exists an integerv 0 such thatA, B v generate a free subgroup ofG for allv ≳=v 0. A suitable value forv 0 is easily calculated. The result can be generalized by replacingW u byWW σ whereW σ is obtained fromW by means of applying certain automorphisms of period 2 toW. Here it is required that the wordWW σ satisfies an easily verifiable extra condition.

1. Notation and preliminary. We shall use the terminology and results in [1]. Let G be a connected semi-simple algebraic group defined over an algebraically closed field K, T a maximal torus of G, B a Borel subgroup containing T, Bu the unipotent part of B, N the normalizer of T in G, W=N/T the Weyl group, and X=X(T) the character group of T. The character group X is a free abelian group of finite rank n. The Weyl group W acts on X by

In (3), Chevalley constructed, for every field K and every semi-simple Lie algebra g over the complex number field, a group GK(g) by using a system of root vectors Xr which satisfies a certain condition (more precisely, (1.2) in Section 1). The main point of Chevalley's construction lies in the fact that the above-mentioned root vectors furnish a basis of g such that, for every root r, exp(t ad Xr) is represented by a matrix (Aij(t)) whose entries Aij(t) are polynomials in t with integral coefficients.

Dans (6), l'auteur a défini, pour chaque entier n ≥ 0, un groupe fini Gn d'ordre q3(q —1)(q3 + 1), où q = 32n+1. Les groupes G1, G2, . . . sont simples, tandis que est simple.En réalisant G = Gn comme un groupe de permutations, opérant à droite, de l'ensemble de q3 + 1 classes à droite modulo le normalisateur N(P) d'un 3-sous-groupe de Sylow de G, on voit aisément que G satisfait aux conditions (0.1)-(0.4) suivantes :(0.1)G est un groupe de permutations doublement transitif d'un ensemble E de m + 1 lettres, où m ≥ 3;(0.2)si a et b sont deux lettres distinctes quelconques de E, le sous-groupe formé de la totalité des permutations dans G qui laissent invariantes a et b contient une, et une seule, permutation ≠ 1 qui laisse au moins 3 lettres invariantes;(0.3)toute involution dans G laisse au moins 3 lettres invariantes (une involution dans un group est par définition un élément d'ordre 2);(0.4)m est impair.

A conjecture of Chowla on the number of integers a between 1 and p − 1 for which the polynomial x n + x + a becomes irreducible (mod p ) is proved. The same arguments show that x n + x + a can be replaced by more general polynomials.

F. Haimo and E. Schenkman [1] raised the question: Does a nilpotent group G always possess an outer automorphism? The answer is in the affirmative if G is finite and nilpotent of class 2, as is seen from a Schenkman's [I ] stronger result. The object of this note is to show that the answer is also in the affirmative for another family of nilpotent groups, namely the family of all finite p-groups G of order greater than 2 such that xi = e for every element x in G. Actually, our result is somewhat stronger

Introduction. A nil group is an (additively written) abelian group G such that every associative ring R with G as its additive group has trivial multiplication, i.e., xy = 0 for every x, y ER. Szele [2] has shown that every nil group is either a torsion group (every element of finite order) or a torsion-free group (every element except the identity of infinite order), and he has completely characterized the torsion nil groups. In [3 ], Szele discusses some questions related to nil groups, but is unable to find conditions for the existence of torsionfree nil groups. An additive group G is said to be strongly nil if there is no nontrivial ring, associative or nonassociative, which has G as its additive group. It is the purpose of this note to characterize those nil and strongly nil groups which are (isomorphic to) a weak direct sum of subgroups of the additive group of rationals, and to show that every such nil group is also strongly nil. The authors would like to express thanks to the referee for helpful suggestions.

Let ƣ be a commutative associative algebra over a field Φ of characteristic p > 0 , and ϑ(ƣ) the derivation algebra of ƣ. A subalgebra [symbol omitted] of ϑ(ƣ) is called regular if f D ɛ for any f ɛ ƣ and D ɛ [symbol omitted]. For a regular subalgebra [symbol omitted] of ϑ(ƣ), if there exist D₁,…, D[subscript]m ɛ [symbol omitted] such that every D ɛ [symbol omitted] is expressed uniquely as D = f₁D₁+ ... + f[subscript]mD[subscript]m, where e f[subscript]i ɛ ƣ, then [symbol omitted] is said to be defined by the system {D₁, …, D[subscript]m} and is denoted by the notation [symbol omitted] (ƣ ; D₁, ..., D[subscript]m). In this dissertation, the family of [symbol omitted] Lie algebras of the type [symbol omitted]( ƣ; D₁, …, D[subscript]m) is studied. It is shown that if ƣ is a field then all algebras in [symbol omitted] are simple except when p = 2, m = 1. It is also shown that if Φ is algebraically closed then every simple algebra in [symbol omitted] is a generalized Witt algebra of the type defined by I. Kaplansky [Bull. Amer. Math. Soc. vol. 10 (1943), pp. 107-121], and, conversely, that every generalized Witt algebra belongs to [symbol omitted]. A simpler form of the generalized Witt algebras is given. By using this form, the problem of whether every generalized Witt algebra can be defined over GF(p) is partly solved. It is shown also that a subfamily [symbol omitted] of [symbol omitted] , consisting for the most part of non-simple algebras, has a remarkable property: every algebra in [symbol omitted] has the same ideal theory as that of a commutative associative algebra. Jacobson's result on automorphisms of the derivation algebras of the group algebras of commutative groups of the type (p, ..., p) is extended to generalized Witt algebras, and, finally, it is shown that m is an invariant of the algebra [symbol omitted] = [symbol omitted] (ƣ; D₁, ..., D[subscript]m) if [symbol omitted] is normal simple.

Out of the classical theory of Riemann surfaces, we extract the following combinatorial theorem: If π 1 ,…, π m are permutations of 1, 2,…, n , such that π 1 … π m = 1, then v ( π 1 ) + … + v ( π m ) ⩾ 2( n − t ), where v ( π ) = n − r , r being the number of orbits of the cyclic group generated by the permutation π acting on the set {1,…, n }, and where t is the number of orbits of the group generated by π 1 ,…, π m . The direct proof of this theorem seems to be difficult.