Cadet Rick Adderly yells at Pennhall and tells him to take off his hat. In any military school or in the military they would never call it a hat, they would call it a cover.
When Penhall arrives at the academy, Cadet Sgt. Major Jackson introduces himself as the company commander. Later in the episode, Cadet Lieutenant Rick Adderly is introduced as the battalion commander. Both of these are incorrect. At a military academy, a company is commanded by a Cadet Captain and a battalion is commanded by the highest ranking cadet at the academy, usually a Cadet Major or Cadet Lieutenant Colonel. A Cadet Lieutenant would command a platoon.