64 research outputs found
The orders of nonsingular derivations of Lie algebras of characteristic two
Nonsingular derivations of modular Lie algebras which have finite
multiplicative order play a role in the coclass theory for pro- groups and
Lie algebras. A study of the set N_p of positive integers which occur as orders
of nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of
positive characteristic p was initiated by Shalev and continued by the present
author. In this paper we continue this study in the case of characteristic two.
Among other results, we prove that any divisor n of 2^k-1 with
belongs to N_2. Our methods consist of elementary arguments
with polynomials over finite fields and a little character theory of finite
groups.Comment: 11 page
Exponential functions in prime characteristic
In this note we determine all power series F(X)\in 1+X\F_p[[X]] such that
has only terms of total degree a multiple of . Up
to a scalar factor, they are all the series of the form for some c\in\F_p and G(X)\in 1+X\F_p[[X]], where
is the Artin-Hasse
exponential.Comment: 6 pages, to be published in Aequationes Mat
Root multiplicities and number of nonzero coefficients of a polynomial
It is known that the weight (that is, the number of nonzero coefficients) of
a univariate polynomial over a field of characteristic zero is larger than the
multiplicity of any of its nonzero roots. We extend this result to an
appropriate statement in positive characteristic. Furthermore, we present a new
proof of the original result, which produces also the exact number of monic
polynomials of a given degree for which the bound is attained. A similar
argument allows us to determine the number of monic polynomials of a given
degree, multiplicity of a given nonzero root, and number of nonzero
coefficients, over a finite field of characteristic larger than the degree.Comment: 6 pages. Minor change from previous version: added Example 6,
illustrating the difficulties arising when one tries to relax the hypothesis
n<p of Theorem
Nottingham Lie algebras with diamonds of finite and infinite type
We consider a class of infinite-dimensional, modular, graded Lie algebras,
which includes the graded Lie algebra associated to the Nottingham group with
respect to its lower central series. We identify two subclasses of Nottingham
Lie algebras as loop algebras of finite-dimensional simple Lie algebras of
Hamiltonian Cartan type. A property of Laguerre polynomials of derivations,
which is related to toral switching, plays a crucial role in our constructions.Comment: 17 pages; minor changes from the previous versio
Automorphisms of p-groups of maximal class
Juhasz has proved that the automorphism group of a group G of maximal class
of order p^n, with p\ge 5 and n>p+1, has order divisible by
. We show that by translating the problem in terms
of derivations, the result can be deduced from the case where G is metabelian.
Here one can use a general result of Caranti and Scoppola concerning
automorphisms of two-generator, nilpotent metabelian groups.Comment: 8 page
- …