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

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 $G$ be an abelian group, and $F$ a downward directed family of subsets of $G$. The finest topology $\mathcal{T}$ on $G$ under which $F$ converges to $0$ has been described by I.Protasov and E.Zelenyuk. In particular, their description yields a criterion for $\mathcal{T}$ to be Hausdorff. They then show that if $F$ is the filter of cofinite subsets of a countable subset $X\subseteq G$, there is a simpler criterion: $\mathcal{T}$ is Hausdorff if and only if for every $g\in G-\{0\}$ and positive integer $n$, there is an $S\in F$ such that $g$ does not lie in the n-fold sum $n(S\cup\{0\}\cup-S)$. In this note, their proof is adapted to a larger class of families $F$. In particular, if $X$ is any infinite subset of $G$, $\kappa$ any regular infinite cardinal $\leq\mathrm{card}(X)$, and $F$ the set of complements in $X$ of subsets of cardinality $<\kappa$, then the above criterion holds. We then give some negative examples, including a countable downward directed set $F$ of subsets of $\mathbb{Z}$ not of the above sort which satisfies the "$g\notin n(S\cup\{0\}\cup-S)$" condition, but does not induce a Hausdorff topology. We end with a version of our main result for noncommutative $G$.Comment: 10 pages. Copy at http://math.berkeley.edu/~gbergman/papers may be updated more frequently than arXiv cop

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

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

Adjoining a universal inner inverse to a ring element

Let $R$ be an associative unital algebra over a field $k,$ let $p$ be an element of $R,$ and let $R'=R\langle q\mid pqp= p\rangle.$ We obtain normal forms for elements of $R',$ and for elements of $R'$-modules arising by extension of scalars from $R$-modules. The details depend on where in the chain $pR\cap Rp \subseteq pR\cup Rp \subseteq pR + Rp \subseteq R$ the unit $1$ of $R$ first appears. This investigation is motivated by a hoped-for application to the study of the possible forms of the monoid of isomorphism classes of finitely generated projective modules over a von Neumann regular ring; but that goal remains distant. We end with a normal form result for the algebra obtained by tying together a $k$-algebra $R$ given with a nonzero element $p$ satisfying $1\notin pR+Rp$ and a $k$-algebra $S$ given with a nonzero $q$ satisfying $1\notin qS+Sq,$ via the pair of relations $p=pqp,$ $q=qpq.$Comment: 28 pages. Results on mutual inner inverses added at end of earlier version, and much clarification of wording etc.. After publication, any updates, errata, related references etc. found will be recorded at http://math.berkeley.edu/~gbergman/paper

Some questions for possible submission to the next Kourovka notebook

This is a collection of questions that I am considering submitting to the next edition of the Kourovka Notebook of open questions in group theory. Most are questions I raised in papers between 1981 and the present; a few are new. I welcome feedback.Comment: 6 pages. Copy at http://math.berkeley.edu/~gbergman/papers/unpub may be updated more frequently than arXiv cop