What is the role of the user ID (UID) and group ID (GID) in the 'mkdir' permission error?

UID and GID determine the user and group associated with a file or directory. The `mkdir` command checks if the user executing the command has write permissions for the parent directory *and* if the file system permissions allow creation of new directories within that parent. Incorrect UID/GID assignments can lead to permission denied errors, especially when working with shared file systems.