If you have an infinite set X of cardinality k, then what is the cardinality of Sym(X) - the group of permutations of X ?



Cardinality of the permutations of an infinite set
$k^k$.


Easy that it's an upper bound. For lower bound split $X$ into two equinumerous
subsets; there are $\ge k^k$ permutations swapping the two subsets.


