8,357 research outputs found
A geometric approach to Quillen's conjecture
We introduce {\em admissible collections} for a finite group and use them
to prove that most of the finite classical groups in non-defining
characteristic satisfy the {\em Quillen dimension at property}, a strong
version of Quillen's conjecture, at a given odd prime divisor of .
Compared to the methods in \cite{AS1993}, our techniques are simpler.Comment: First Author supported by MEC grant MTM2016-78647-P and Junta de
Andaluc\'ia grant FQM-21
Dense normal subgroups and chief factors in locally compact groups
In 'The essentially chief series of a compactly generated locally compact
group', an analogue of chief series for finite groups is discovered for
compactly generated locally compact groups. In the present article, we show
that chief factors necessarily exist in all locally compact groups with
sufficiently rich topological structure. We also show that chief factors have
one of seven types, and for all but one of these types, there is a
decomposition into discrete groups, compact groups, and topologically simple
groups.
Our results for chief factors require exploring the theory developed in
'Chief factors in Polish groups' in the setting of locally compact groups. In
this context, we obtain tighter restrictions on the factorization of normal
compressions and the structure of quasi-products. Consequently, both
(non-)amenability and elementary decomposition rank are preserved by normal
compressions
The Cayley isomorphism property for the group
A finite group is called a DCI-group if two Cayley digraphs over are
isomorphic if and only if their connection sets are conjugate by a group
automorphism. We prove that the group , where is a prime,
is a DCI-group if and only if . Together with the previously obtained
results, this implies that a group of order , where is a prime, is
a DCI-group if and only if and .Comment: 19 pages. arXiv admin note: text overlap with arXiv:2003.08118,
arXiv:1912.0883
Elementary Abelian p-groups of rank 2p+3 are not CI-groups
For every prime we exhibit a Cayley graph of
which is not a CI-graph. This proves that an elementary Abelian -group of
rank greater than or equal to is not a CI-group. The proof is elementary
and uses only multivariate polynomials and basic tools of linear algebra.
Moreover, we apply our technique to give a uniform explanation for the recent
works concerning the bound.Comment: 11 page
The Cayley isomorphism property for Cayley maps
In this paper we study finite groups which have Cayley isomorphism property
with respect to Cayley maps, CIM-groups for a brief. We show that the structure
of the CIM-groups is very restricted. It is described in Theorem~\ref{111015a}
where a short list of possible candidates for CIM-groups is given.
Theorem~\ref{111015c} provides concrete examples of infinite series of
CIM-groups
On the Cohomology of Central Frattini Extensions
We use topological methods to compute the mod p cohomology of certain
p-groups. More precisely we look at central Frattini extensions of elementary
abelian by elementary abelian groups such that their defining k-invariants span
the entire image of the Bockstein. We show that if p is sufficiently large,
then the mod p cohomology of the extension can be explicitly computed as an
algebra
C-loops: extensions and constructions
C-loops are loops satisfying the identity . We
develop the theory of extensions of C-loops, and characterize all nuclear
extensions provided the nucleus is an abelian group. C-loops with central
squares have very transparent extensions; they can be built from small blocks
arising from the underlying Steiner triple system. Using these extensions, we
decide for which abelian groups and Steiner loops there is a
nonflexible C-loop with center such that is isomorphic to . We
discuss possible orders of associators in C-loops. Finally, we show that the
loops of signed basis elements in the standard real Cayley-Dickson algebras are
C-loops.Comment: 17 pages, amsar
A semigroup-theoretical view of direct-sum decompositions and associated combinatorial problems
Let be a ring and let be a small class of right -modules
which is closed under finite direct sums, direct summands, and isomorphisms.
Let denote a set of representatives of isomorphism
classes in and, for any module in , let
denote the unique element in isomorphic to . Then
is a reduced commutative semigroup with operation
defined by , and this semigroup carries all
information about direct-sum decompositions of modules in . This
semigroup-theoretical point of view has been prevalent in the theory of
direct-sum decompositions since it was shown that if
is semilocal for all , then is a
Krull monoid. Suppose that the monoid is Krull with a
finitely generated class group (for example, when is the class of
finitely generated torsion-free modules and is a one-dimensional reduced
Noetherian local ring). In this case we study the arithmetic of using new methods from zero-sum theory. Furthermore, based on
module-theoretic work of Lam, Levy, Robson, and others we study the algebraic
and arithmetic structure of the monoid for certain
classes of modules over Pr\"ufer rings and hereditary Noetherian prime rings.Comment: 42 pages; to appear in the Journal of Algebra and its Application
Class 2 Moufang loops, small Frattini Moufang loops, and code loops
Let be a Moufang loop which is centrally nilpotent of class 2. We first
show that the nuclearly-derived subloop (normal associator subloop) of
has exponent dividing 6. It follows that (the subloop of of
elements of -power order) is associative for . Next, a loop is said
to be a {\it small Frattini Moufang loop}, or SFML, if has a central
subgroup of order such that C\isom L/Z is an elementary abelian
-group. is thus given the structure of what we call a {\it coded vector
space}, or CVS. (In the associative/group case, CVS's are either orthogonal
spaces, for , or symplectic spaces with attached linear forms, for .)
Our principal result is that every CVS may be obtained from an SFML in this
way, and two SFML's are isomorphic in a manner preserving the central subgroup
if and only if their CVS's are isomorphic up to scalar multiple.
Consequently, we obtain the fact that every SFM 2-loop is a code loop, in the
sense of Griess, and we also obtain a relatively explicit characterization of
isotopy in SFM 3-loops. (This characterization of isotopy is easily extended to
Moufang loops of class 2 and exponent 3.) Finally, we sketch a method for
constructing any finite Moufang loop which is centrally nilpotent of class 2
Essential dimensions of algebraic groups and a resolution theorem for G-varieties
Let G be an algebraic group and let X be a generically free G-variety. We
show that X can be transformed, by a sequence of blowups with smooth
G-equivariant centers, into a G-variety X' with the following property: the
stabilizer of every point of X' is isomorphic to a semidirect product of a
unipotent group U and a diagonalizable group A.
As an application of this and related results, we prove new lower bounds on
essential dimensions of some algebraic groups. We also show that certain
polynomials in one variable cannot be simplified by a Tschirnhaus
transformation.Comment: This revision contains new lower bounds for essential dimensions of
algebraic groups of types A_n and E_7. AMS LaTeX 1.1, 42 pages. Paper by
Zinovy Reichstein and Boris Youssi, includes an appendix by J\'anos Koll\'ar
and Endre Szab\'o. Author-supplied dvi file available at
http://ucs.orst.edu/~reichstz/pub.htm
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