214 research outputs found
Normalizers of Primitive Permutation Groups
Let be a transitive normal subgroup of a permutation group of finite
degree . The factor group can be considered as a certain Galois group
and one would like to bound its size. One of the results of the paper is that
if is primitive unless , , , , or
. This bound is sharp when is prime. In fact, when is
primitive, unless is a member of a given infinite
sequence of primitive groups and is different from the previously listed
integers. Many other results of this flavor are established not only for
permutation groups but also for linear groups and Galois groups.Comment: 44 pages, grant numbers updated, referee's comments include
Computations for Coxeter arrangements and Solomon's descent algebra II: Groups of rank five and six
In recent papers we have refined a conjecture of Lehrer and Solomon
expressing the character of a finite Coxeter group acting on the th
graded component of its Orlik-Solomon algebra as a sum of characters induced
from linear characters of centralizers of elements of . Our refined
conjecture relates the character above to a component of a decomposition of the
regular character of related to Solomon's descent algebra of . The
refined conjecture has been proved for symmetric and dihedral groups, as well
as finite Coxeter groups of rank three and four.
In this paper, the second in a series of three dealing with groups of rank up
to eight (and in particular, all exceptional Coxeter groups), we prove the
conjecture for finite Coxeter groups of rank five and six, further developing
the algorithmic tools described in the previous article. The techniques
developed and implemented in this paper provide previously unknown
decompositions of the regular and Orlik-Solomon characters of the groups
considered.Comment: Final Version. 17 page
Algebraic curves with many automorphisms
Let be a (projective, geometrically irreducible, nonsingular) algebraic
curve of genus defined over an algebraically closed field of odd
characteristic . Let be the group of all automorphisms of which
fix element-wise. It is known that if then the -rank
(equivalently, the Hasse-Witt invariant) of is zero. This raises the
problem of determining the (minimum-value) function such that whenever
then has zero -rank. For {\em{even}} we prove
that . The {\em{odd}} genus case appears to be much more
difficult although, for any genus , if has a solvable
subgroup such that then has zero -rank and fixes a
point of . Our proofs use the Hurwitz genus formula and the Deuring
Shafarevich formula together with a few deep results from finite group theory
characterizing finite simple groups whose Sylow -subgroups have a cyclic
subgroup of index . We also point out some connections with the Abhyankar
conjecture and the Katz-Gabber covers
Products of conjugacy classes and fixed point spaces
We prove several results on products of conjugacy classes in finite simple
groups. The first result is that there always exists a uniform generating
triple. This result and other ideas are used to solve a 1966 conjecture of
Peter Neumann about the existence of elements in an irreducible linear group
with small fixed space. We also show that there always exist two conjugacy
classes in a finite non-abelian simple group whose product contains every
nontrivial element of the group. We use this to show that every element in a
non-abelian finite simple group can be written as a product of two rth powers
for any prime power r (in particular, a product of two squares).Comment: 44 page
Recommended from our members
Computational Group Theory
This was the seventh workshop on Computational Group Theory. It showed that Computational Group Theory has significantly expanded its range of activities. For example, symbolic computations with groups and their representations and computations with infinite groups play a major role nowadays. The talks also presented connections and applications to cryptography, number theory and the algorithmic theory of algebras
Using the Maple Computer Algebra System as a Tool for Studying Group Theory
The purpose of this study was to show that computers can be powerful tools for studying group theory. Specifically the author examined ways that the computer algebra system Maple can be used to assist in the study of group theory. The study consists of four main parts.
After a brief introduction in chapter one, chapter two discusses simple procedures written by the author to study small finite groups. These procedures rely on the fact that for small finite groups, the elements can all be stored on a computer and tested for various properties. All of the procedures are contained in the appendix, and each is described in chapter two.
The Maple software comes with a built in set of group theory procedures. The procedures work with two types of groups, permutation groups and finitely presented groups. The author discusses all of the procedures dealing with permutation groups in chapter three and the procedures for finitely presented groups in chapter four. The main theoretical tool for permutation groups is a stabilizer chain, and the main tool for finitely presented groups is the Todd-Coxeter algorithm. Both of these methods and their implementations in Maple are discussed in detail.
The study is concluded by examining some applications of group theory. The author discusses check digit schemes, RSA encryption, and permutation factoring. The ability to factor a permutation in terms of a set of generators can be used to solve several puzzles such as the Rubik\u27s cube
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