  We study the group $T_A$ of rearrangements of the Airplane limit space
introduced by Belk and Forrest in [3]. We prove that $T_A$ is generated by a
copy of Thompson's group $F$ and a copy of Thompson's group $T$, hence it is
finitely generated. Then we study the commutator subgroup $[T_A, T_A]$, proving
that the abelianization of $T_A$ is isomorphic to $\mathbb{Z}$ and that $[T_A,
T_A]$ is simple, finitely generated and acts 2-transitively on the so-called
components of the Airplane limit space. Moreover, we show that $T_A$ is
contained in $T$ and contains a natural copy of the Basilica rearrangement
group $T_B$ studied in [2].
