The cardinality of the permutations on a set with finite non-fixed points

Abstract

In the Zermelo-Frankel set theory (ZF) with the Axiom of Choice (AC), the set of subsets of X, P(X), and the set of permutations on X, S(X), have the same cardinality for any imfinite set X. Dawson and Howard showed that without AC, we cannot conclude any relationship between these cardinals. Halbeisen and Shelah showed, in ZF, that in(X)| < (X)| for any infinite set X, where fin(X) is the set of finite subsets of X. With AC, in(X)| = fin(X)| for any infinite set X, where Sfin(X) is the set of permutations on X with finite non-fixed points. However, in contrast with the relation between in(X)| and (X)|, Tachtsis showed that fin(X)| ≠ (X) is not provable in ZF for an arbitrary infinite set X. In this project, we study relationship between in(X)| and fin(X)| for an infinite set X in the absence of AC and give some conditions that make them comparable.