16,427 research outputs found
Generating infinite symmetric groups
Let S=Sym(\Omega) be the group of all permutations of an infinite set \Omega.
Extending an argument of Macpherson and Neumann, it is shown that if U is a
generating set for S as a group, respectively as a monoid, then there exists a
positive integer n such that every element of S may be written as a group word,
respectively a monoid word, of length \leq n in the elements of U.
Several related questions are noted, and a brief proof is given of a result
of Ore's on commutators that is used in the proof of the above result.Comment: 9 pages. See also http://math.berkeley.edu/~gbergman/papers To
appear, J.London Math. Soc.. Main results as in original version. Starting on
p.4 there are references to new results of others including an answer to
original Question 8; "sketch of proof" of Lemma 11 is replaced by a full
proof; 6 new reference
Critical point for the strong field magnetoresistance of a normal conductor/perfect insulator/perfect conductor composite with a random columnar microstructure
A recently developed self-consistent effective medium approximation, for
composites with a columnar microstructure, is applied to such a
three-constituent mixture of isotropic normal conductor, perfect insulator, and
perfect conductor, where a strong magnetic field {\bf B} is present in the
plane perpendicular to the columnar axis. When the insulating and perfectly
conducting constituents do not percolate in that plane, the
microstructure-induced in-plane magnetoresistance is found to saturate for
large {\bf B}, if the volume fraction of the perfect conductor is greater
than that of the perfect insulator . By contrast, if , that
magnetoresistance keeps increasing as without ever saturating. This
abrupt change in the macroscopic response, which occurs when , is a
critical point, with the associated critical exponents and scaling behavior
that are characteristic of such points. The physical reasons for the singular
behavior of the macroscopic response are discussed. A new type of percolation
process is apparently involved in this phenomenon.Comment: 4 pages, 1 figur
Direct limits and fixed point sets
For which groups G is it true that whenever we form a direct limit of G-sets,
dirlim_{i\in I} X_i, the set of its fixed points, (dirlim_I X_i)^G, can be
obtained as the direct limit dirlim_I(X_i^G) of the fixed point sets of the
given G-sets? An easy argument shows that this holds if and only if G is
finitely generated.
If we replace ``group G'' by ``monoid M'', the answer is the less familiar
condition that the improper left congruence on M be finitely generated.
Replacing our group or monoid with a small category E, the concept of set on
which G or M acts with that of a functor E --> Set, and the concept of fixed
point set with that of the limit of a functor, a criterion of a similar nature
is obtained. The case where E is a partially ordered set leads to a condition
on partially ordered sets which I have not seen before (pp.23-24, Def. 12 and
Lemma 13).
If one allows the {\em codomain} category Set to be replaced with other
categories, and/or allows direct limits to be replaced with other kinds of
colimits, one gets a vast area for further investigation.Comment: 28 pages. Notes on 1 Aug.'05 revision: Introduction added; Cor.s 9
and 10 strengthened and Cor.10 added; section 9 removed and section 8
rewritten; source file re-formatted for Elsevier macros. To appear, J.Al
On group topologies determined by families of sets
Let be an abelian group, and a downward directed family of subsets of
. The finest topology on under which converges to
has been described by I.Protasov and E.Zelenyuk. In particular, their
description yields a criterion for to be Hausdorff. They then
show that if is the filter of cofinite subsets of a countable subset
, there is a simpler criterion: is Hausdorff if and
only if for every and positive integer , there is an
such that does not lie in the n-fold sum .
In this note, their proof is adapted to a larger class of families . In
particular, if is any infinite subset of , any regular infinite
cardinal , and the set of complements in of
subsets of cardinality , then the above criterion holds.
We then give some negative examples, including a countable downward directed
set of subsets of not of the above sort which satisfies the
"" condition, but does not induce a Hausdorff
topology.
We end with a version of our main result for noncommutative .Comment: 10 pages. Copy at http://math.berkeley.edu/~gbergman/papers may be
updated more frequently than arXiv cop
Can one factor the classical adjoint of a generic matrix?
Let k be a field, n a positive integer, X a generic nxn matrix over k (i.e.,
a matrix (x_{ij}) of n^2 independent indeterminates over the polynomial ring
k[x_{ij}]), and adj(X) its classical adjoint. It is shown that if char k=0 and
n is odd, then adj(X) is not the product of two noninvertible nxn matrices over
k[x_{ij}]. If n is even and >2, a restricted class of nontrivial factorizations
occur. The nonzero-characteristic case remains open.
The operation adj on matrices arises from the (n-1)st exterior power functor
on modules; the same question can be posed for matrix operations arising from
other functors.Comment: Revised version contains answer to "even n" question left open in
original version. (Answer due to Buchweitz & Leuschke; simple proof in this
note.) Copy at http://math.berkeley.edu/~gbergman/papers will always have
latest version; revisions sent to arXiv only for major change
More Abelian groups with free duals
In answer to a question of A. Blass, J. Irwin and G. Schlitt, a subgroup G of
the additive group Z^{\omega} is constructed whose dual, Hom(G,Z), is free
abelian of rank 2^{\aleph_0}. The question of whether Z^{\omega} has subgroups
whose duals are free of still larger rank is discussed, and some further
classes of subgroups of Z^{\omega} are noted.Comment: 9 pages. Copy at http://math.berkeley.edu/~gbergman/papers may be
updated more frequently than arXiv cop
An inner automorphism is only an inner automorphism, but an inner endomorphism can be something strange
The inner automorphisms of a group G can be characterized within the category
of groups without reference to group elements: they are precisely those
automorphisms of G that can be extended, in a functorial manner, to all groups
H given with homomorphisms G --> H. Unlike the group of inner automorphisms of
G itself, the group of such extended systems of automorphisms is always
isomorphic to G. A similar characterization holds for inner automorphisms of an
associative algebra R over a field K; here the group of functorial systems of
automorphisms is isomorphic to the group of units of R modulo units of K.
If one substitutes "endomorphism" for "automorphism" in these considerations,
then in the group case, the only additional example is the trivial
endomorphism; but in the K-algebra case, a construction unfamiliar to ring
theorists, but known to functional analysts, also arises.
Systems of endomorphisms with the same functoriality property are examined in
some other categories; other uses of the phrase "inner endomorphism" in the
literature, some overlapping the one introduced here, are noted; the concept of
an inner {\em derivation} of an associative or Lie algebra is looked at from
the same point of view, and the dual concept of a "co-inner" endomorphism is
briefly examined. Several questions are posed.Comment: 20 pages. To appear, Publicacions Mathem\`{a}tiques. The 1-1-ness
result in the appendix has been greatly strengthened, an "Overview" has been
added at the beginning, and numerous small rewordings have been made
throughou
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