Almost disjoint permutation groups. (English) Zbl 0854.20003
It is shown that if \(\delta\), \(\kappa\), \(\lambda\) are uncountable cardinals with \(\delta\leq\kappa\leq\lambda\), then the following are equivalent: (i) there is a subgroup \(G\) of \(\text{Sym}(\kappa)\) of cardinality \(\lambda\) in which each non-identity element fixes less than \(\delta\) points; (ii) there is a family of size \(\lambda\) of \(\kappa\)-element subsets of \(\kappa\) with pairwise intersection of cardinality less then \(\delta\).
Furthermore, given infinite cardinals \(\kappa\) and \(\lambda\), the following are equivalent: (i) there is a subgroup \(G\) of \(\text{Sym}(\kappa)\) of cardinality \(\lambda\) in which each non-identity element fixes less than \(\kappa\) points; (ii) there is a family of size \(\lambda\) of \(\kappa\)-element subsets of \(\kappa\) with pairwise intersection of cardinality less than \(\kappa\).
Furthermore, given infinite cardinals \(\kappa\) and \(\lambda\), the following are equivalent: (i) there is a subgroup \(G\) of \(\text{Sym}(\kappa)\) of cardinality \(\lambda\) in which each non-identity element fixes less than \(\kappa\) points; (ii) there is a family of size \(\lambda\) of \(\kappa\)-element subsets of \(\kappa\) with pairwise intersection of cardinality less than \(\kappa\).
Reviewer: H.D.Macpherson (Leeds)
References:
[1] | James E. Baumgartner, Almost-disjoint sets, the dense set problem and the partition calculus, Ann. Math. Logic 9 (1976), no. 4, 401 – 439. · Zbl 0339.04003 · doi:10.1016/0003-4843(76)90018-8 |
[2] | Fred Galvin, Generating countable sets of permutations, J. London Math. Soc. (2) 51 (1995), 230–242. CMP 95:10 · Zbl 0837.20005 |
[3] | W. Sierpinski, Sur une décomposition d’ensembles, Monatsh. Math. Phys. 35 (1928), 239–242. · JFM 54.0092.01 |
[4] | Neil H. Williams, Combinatorial set theory, North-Holland, Amsterdam, 1977. · Zbl 0362.04008 |
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