Permutation/Partition of ground set/Subgroup/Exercise

Let ${}M$ be a set, and let ${}M=\biguplus _{i\in I}M_{i}$ be a partition of ${}M$, that is, every ${}M_{i}$ is a subset of ${}M$, and ${}M$ is the disjoint union of the ${}M_{i}$. Show that the product group

$\prod _{i\in I}\operatorname {Perm} \,(M_{i})$

is a subgroup of ${}\operatorname {Perm} \,(M)$.

Create a solution