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The finite subsets and the permutations with finitely many non‐fixed points of a set

We write fin(m) and Sfin(m) for the cardinalities of the set of finite subsets and the set of permutations with finitely many non‐fixed points, respectively, of a set which is of cardinality m . In this paper, we investigate relationships between fin(m) and Sfin(m) for an infinite cardinal m in the absence of the Axiom of Choice. We give conditions that make fin(m) and Sfin(m) comparable as well as give related consistency results.