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

Abstract

In this thesis, we study relationships between |Sn(A)| and |seqn1-1(A)| as well as |seqn(A)| for infinite sets A, where Sn(A) is the set of permutations of A with n non-fixed points and seqn(A) and seqn1-1(A) are the set of sequences and the set of one-to-one sequences of elements of A with length n, respectively, where n is a natural number greater than 1.

With the Axiom of Choice (AC), |Sn(A)|, |seqn1-1(A)|, and |seqn(A)| are equal for all infinite sets A. Among our results, we show, in the Zermelo-Fraenkel set theory (ZF), that |Sn(A)|<=|seqn1-1(A)| for any infinite set A under some weak form of AC and the assumption cannot be removed. In the other direction, we show that |seqn1-1(A)|<=|Sn+1(A)| for any infinite set A and the subscript n+1 cannot be reduced to n. Moreover, we also show that "|Sn(A)|<=|Sn+1(A)| for any infinite set A" is not provable in ZF.