Companies, Troops, and Batteries (different designations of the same level units) are comprised of 60-200 soldiers, assigned to 3-5 platoons. Companies are arranged in groups of 4-6 in a Battalion under the command of a Lieutenant Colonel. Battalions are arranged in groups of three or more under Brigades, Groups, or Regiments (depending on the branch) and are commanded by a Colonel or Brigadier General. Brigades/Groups/Regiments are supported by Artillery, Engineer, and Combat Support Battalions.
Brigades are arranged in groups of three at the Division level. There are five types of divisions: Light Infantry, Mechanized Infantry, Armor, Airborne, and Air Assault. Divisions are comprised of 10,000 to 16,000 soldiers. The smallest number of Companies assigned to a Divisions would then be twenty-seven (27), but the actual number of Companies/Troops/Batteries in any of the US Army's Divisions is higher, considering deployment and support assignments.
Many US Army divisions have nicknames. The US Army's 101st Airborne division (which is the division trained for air assault operations) is nicknamed 'The Screaming Eagles'.
between 10,000 and 20,000 soldiers
The 82nd Airborne Division was a paratrooper division and therefore it was smaller than the typical Infantry Division. A paratrooper division were authorized to have about 9,000 men in its organization whereas an Infantry Division had 15,000 men. An airborne division had 9 companies in each of the three regiments whereas the infantry division had 12 companies in each of its regiments.
That depends on which division you're referring to - there are several divisons in the US Army, US Army Reserve, and US Army National Guard. The rank typically held by a Division Commander is Major General (two stars).
US ARMY Army Group Army Corps Division Regiment Battalion Company Platoon Squad
The Army Corps of Engineers
In WWII it belonged to the 26th Yankee Division. Part of the 3 rd Army.
The US Army's 1st Cavalry Division .
how many army airfileds are in the US
Not a conventional US Army division. But the US Army's 75th Rangers were in South Vietnam. The 75th had "Ranger" companies strung out in each MR (Military Region I, II, III, and IV/also called I Corps, II Corps, etc.). For example P company (called Papa 75th Rangers) were in I Corps. Kilo 75th (K company) may have been in either II or III Corps, etc.
15th Infantry regiment, 3rd Infantry Division (US Army) and 36th Infantry Division (Texas National Guard)
The US 9th Armored Division captured the bridge at Remagen. The US 27th Armored Infantry Battalion was the first group to cross it. It was called "Operation Lumberjack".