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

On semilinear representations of the infinite symmetric group

In this note the smooth (i.e. with open stabilizers) linear and {\sl semilinear} representations of certain permutation groups (such as infinite symmetric group or automorphism group of an infinite-dimensional vector space over a finite field) are studied. Many results here are well-known to the experts, at least in the case of {\sl linear representations} of symmetric group. The presented results suggest, in particular, that an analogue of Hilbert's Theorem 90 should hold: in the case of faithful action of the group on the base field the irreducible smooth semilinear representations are one-dimensional (and trivial in appropriate sense).Comment: 19 pages, significant changes; an analogue of Hilbert's Theorem 90 for infinite symmetric groups moved to arXiv:1508.0226

Equivariant properties of symmetric products

The filtration on the infinite symmetric product of spheres by the number of factors provides a sequence of spectra between the sphere spectrum and the integral Eilenberg-Mac Lane spectrum. This filtration has received a lot of attention and the subquotients are interesting stable homotopy types. While the symmetric product filtration has been a major focus of research since the 1980s, essentially nothing was known when one adds group actions into the picture. We investigate the equivariant stable homotopy types, for compact Lie groups, obtained from this filtration of infinite symmetric products of representation spheres. The situation differs from the non-equivariant case, for example the subquotients of the filtration are no longer rationally trivial and on the zeroth equivariant homotopy groups an interesting filtration of the augmentation ideals of the Burnside rings arises. Our method is by global homotopy theory, i.e., we study the simultaneous behavior for all compact Lie groups at once.Comment: 33 page