Commutative decomposition of infinite symmetric groups and transformation monoids

The commutative subgroup width of a group $G$ is the smallest $k$ such that there are abelian subgroups $A_0,A_1,\ldots,A_{k-1}\leq G$ with $G=A_0A_1\cdots A_{k-1}$. Commutative (inverse) submonoid width is defined analogously. In 2002, Ab\'{e}rt showed, rather surprisingly, that the commutative subgroup width of the symmetric group on an infinite set is always finite. It was later shown by Seress that it is always bounded above by $14$. We answer a question of Seress and show that in fact the commutative subgroup width of $\operatorname{Sym}(\mathbb{N})$ is at most $9$. We improve the best known lower bound to $4$. We also study standard monoid analogues of the symmetric group; showing that the commutative submonoid widths of the full transformation monoid $\mathbb{N}^\mathbb{N}$, the partial transformation monoid $P_\mathbb{N}$ and the symmetric inverse monoid $I_\mathbb{N}$ are exactly $3$. We conclude by showing that the commutative inverse submonoid width of any infinite symmetric inverse monoid is always infinite.