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

Closed subgroups of the infinite symmetric group

Let S=Sym(\Omega) be the group of all permutations of a countably infinite set \Omega, and for subgroups G_1, G_2\leq S let us write G_1\approx G_2 if there exists a finite set U\subseteq S such that = . It is shown that the subgroups closed in the function topology on S lie in precisely four equivalence classes under this relation. Which of these classes a closed subgroup G belongs to depends on which of the following statements about pointwise stabilizer subgroups G_{(\Gamma)} of finite subsets \Gamma\subseteq\Omega holds: (i) For every finite set \Gamma, the subgroup G_{(\Gamma)} has at least one infinite orbit in \Omega. (ii) There exist finite sets \Gamma such that all orbits of G_{(\Gamma)} are finite, but none such that the cardinalities of these orbits have a common finite bound. (iii) There exist finite sets \Gamma such that the cardinalities of the orbits of G_{(\Gamma)} have a common finite bound, but none such that G_{(\Gamma)}=\{1\}. (iv) There exist finite sets \Gamma such that G_{(\Gamma)}=\{1\}. Some questions for further investigation are discussed.Comment: 33 pages. See also http://math.berkeley.edu/~gbergman/papers and http://shelah.logic.at (pub. 823). To appear, Alg.Univ., issue honoring W.Taylor. Main results as before (greater length due to AU formatting), but some new results in \S\S11-12. Errors in subscripts between displays (12) and (13) fixed. Error in title of orig. posting fixed. 1 ref. adde

Covering monolithic groups with proper subgroups

Given a finite non-cyclic group $G$, call $\sigma(G)$ the smallest number of proper subgroups of $G$ needed to cover $G$. Lucchini and Detomi conjectured that if a nonabelian group $G$ is such that $\sigma(G) < \sigma(G/N)$ for every non-trivial normal subgroup $N$ of $G$ then $G$ is \textit{monolithic}, meaning that it admits a unique minimal normal subgroup. In this paper we show how this conjecture can be attacked by the direct study of monolithic groups.Comment: I wrote this paper for the Proceedings of the conference "Ischia Group Theory 2012" (March, 26th - 29th 2012

The Effect of an Employer Health Insurance Mandate on Health Insurance Coverage and the Demand for Labor: Evidence from Hawaii

Over the past few decades, policy makers have considered employer mandates as a strategy for stemming the tide of declining health insurance coverage. In this paper we examine the long term effects of the only employer health insurance mandate that has ever been enforced in the United States, Hawaii's Prepaid Health Care Act, using a standard supply-demand framework and Current Population Survey data covering the years 1979 to 2005. During this period, the coverage gap between Hawaii and other states increased, as did real health insurance costs, implying a rising burden of the mandate on Hawaii's employers. We use a variant of the traditional permutation (placebo) test across all states to examine the magnitude and statistical properties of these growing coverage differences and their impacts on labor market outcomes, conditional on an extensive set of covariates. As expected, the coverage gap is larger for workers who tend to have low rates of coverage in the voluntary market (primarily those with lower skills). We also find that relative wages fell in Hawaii over time, but the estimates are statistically insignificant. By contrast, a parallel analysis of workers employed fewer than 20 hours per week indicates that the law significantly increased employers' reliance on such workers in order to reduce the burden of the mandate. We find no evidence suggesting that the law reduced employment probabilities.health insurance, employment, hours, wages

Perturbation theory for normal operators

Let $E \ni x\mapsto A(x)$ be a $\mathscr{C}$-mapping with values unbounded normal operators with common domain of definition and compact resolvent. Here $\mathscr{C}$ stands for $C^\infty$, $C^\omega$ (real analytic), $C^{[M]}$ (Denjoy--Carleman of Beurling or Roumieu type), $C^{0,1}$ (locally Lipschitz), or $C^{k,\alpha}$. The parameter domain $E$ is either $\mathbb R$ or $\mathbb R^n$ or an infinite dimensional convenient vector space. We completely describe the $\mathscr{C}$-dependence on $x$ of the eigenvalues and the eigenvectors of $A(x)$. Thereby we extend previously known results for self-adjoint operators to normal operators, partly improve them, and show that they are best possible. For normal matrices $A(x)$ we obtain partly stronger results.Comment: 32 pages, Remark 7.5 on m-sectorial operators added, accepted for publication in Trans. Amer. Math. So

The combinatorics and the homology of the poset of subgroups of p-power index

AbstractFor a finite group G and a prime p the poset Sp (G) of all subgroups H ≠ G of p-power index is studied. The Möbius number of the poset is given and the homotopy type of the poset is determined as a wedge of spheres. We describe the representation of G on the homology groups of the order complex of Sp (G) and show that this representation can be realized by matrices with entries in the set {+1, -1, 0}. Finally a CL-shellable subposet of Sp (G) is exhibited for odd primes p