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

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

Strong inner inverses in endomorphism rings of vector spaces

For $V$ a vector space over a field, or more generally, over a division ring, it is well-known that every $x\in\mathrm{End}(V)$ has an inner inverse, i.e., an element $y\in\mathrm{End}(V)$ satisfying $xyx=x.$ We show here that a large class of such $x$ have inner inverses $y$ that satisfy with $x$ an infinite family of additional monoid relations, making the monoid generated by $x$ and $y$ what is known as an inverse monoid (definition recalled). We obtain consequences of these relations, and related results. P. Nielsen and J. \v{S}ter, in a paper to appear, show that a much larger class of elements $x$ of rings $R,$ including all elements of von Neumann regular rings, have inner inverses satisfying arbitrarily large finite subsets of the abovementioned set of relations. But we show by example that the endomorphism ring of any infinite-dimensional vector space contains elements having no inner inverse that simultaneously satisfies all those relations. A tangential result proved is a condition on an endomap $x$ of a set $S$ that is necessary and sufficient for $x$ to belong to an inverse submonoid of the monoid of all endomaps of $S.$Comment: 18pp. The main change from the preceding version is the discussion of three questions posed by the referee, two on p.10, starting on line 6, and one starting at the top of p.16. There are also many small revisions of wording et