16,427 research outputs found

    Generating infinite symmetric groups

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    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

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    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 pSp_S is greater than that of the perfect insulator pIp_I. By contrast, if pS<pIp_S<p_I, that magnetoresistance keeps increasing as B2{\bf B}^2 without ever saturating. This abrupt change in the macroscopic response, which occurs when pS=pIp_S=p_I, 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

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    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

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    Let GG be an abelian group, and FF a downward directed family of subsets of GG. The finest topology T\mathcal{T} on GG under which FF converges to 00 has been described by I.Protasov and E.Zelenyuk. In particular, their description yields a criterion for T\mathcal{T} to be Hausdorff. They then show that if FF is the filter of cofinite subsets of a countable subset X⊆GX\subseteq G, there is a simpler criterion: T\mathcal{T} is Hausdorff if and only if for every g∈G−{0}g\in G-\{0\} and positive integer nn, there is an S∈FS\in F such that gg does not lie in the n-fold sum n(S∪{0}∪−S)n(S\cup\{0\}\cup-S). In this note, their proof is adapted to a larger class of families FF. In particular, if XX is any infinite subset of GG, κ\kappa any regular infinite cardinal ≤card(X)\leq\mathrm{card}(X), and FF the set of complements in XX of subsets of cardinality <κ<\kappa, then the above criterion holds. We then give some negative examples, including a countable downward directed set FF of subsets of Z\mathbb{Z} not of the above sort which satisfies the "g∉n(S∪{0}∪−S)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 GG.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?

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    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

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    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

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    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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