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:
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