5 research outputs found

    Unsolved Problems in Group Theory. The Kourovka Notebook

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    This is a collection of open problems in group theory proposed by hundreds of mathematicians from all over the world. It has been published every 2-4 years in Novosibirsk since 1965. This is the 19th edition, which contains 111 new problems and a number of comments on about 1000 problems from the previous editions.Comment: A few new solutions and references have been added or update

    Multicoloured Random Graphs: Constructions and Symmetry

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    This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

    Cardinal invariants related to permutation groups

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    AbstractWe consider the possible cardinalities of the following three cardinal invariants which are related to the permutation group on the set of natural numbers:ag≔ the least cardinal number of maximal cofinitary permutation groups;ap≔ the least cardinal number of maximal almost disjoint permutation families;c(Sym(N))≔ the cofinality of the permutation group on the set of natural numbers.We show that it is consistent with ZFC that ap=ag<c(Sym(N))=ℵ2; in fact we show that in the Miller model ap=ag=ℵ1<ℵ2=c(Sym(N))

    Adjoining cofinitary permutations

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    We show that it is consistent with ZFC + -CH that there is a maximal cofinitary group (or, maximal almost disjoint group) G less than or equal to Sym(omega) such that G is a proper subset of an almost disjoint Family A subset of or equal to sym(omega) and \ G \ < \ A \. We also ask several questions in this area
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