Past exam of the mathematics course of the University of Cambridge / 2020 / ia / Paper 2 / 8D / c / ii

The set $T$ is uncountable. Partition $\mathbb{N}$ into countably many infinite sets $B_1,B_2,\dots$, and choose bijections $h_j:B_j\to \mathbb{Z}$. For every subset $A\subseteq \mathbb{N}$, define a permutation $f_A$ separately on each block byEvery positive iterate translates each integer coordinate by a nonzero amount, so it has no fixed point and $f_A\in T$. Distinct subsets $A$ give distinct permutations, yielding an injection $\mathcal{P}(\mathbb{N})\hookrightarrow T$. Since the power set is uncountable,In fact its cardinality is the continuum, because it is a subset of all functions $\mathbb{N}\to \mathbb{N}$.