the battalion is part of a division. usually there are 3 battalions in a division
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The 1st Infantry Division, the 1st Armored Division and the 509th Parachute Infantry Battalion and the 1st US Ranger Battalion.
The German attack fell mostly on the US First Army. The US Ninth Army was north of the First, and the British were north of the Ninth Army. Because the German breakthrough appeared as though it were going to reach a small Belgian town through which the landlines strung by Signal Corps troops for telephone communications ran, Eisenhower made the decision to place the troops north of the German breakthrough temporarily under the command of the 21st Army Group, commanded by Montgomery, until the troops north and south could cut through the Bulge and reunite - about eight or nine days. This put Montgomery over US troops, and after the battle was stabilized he made some outlandish claims trying to seize credit for having to "rescue" the incompetent Americans. This was par for the course for Montgomery, yet another of his "what a good boy am I" pronouncements, and came very near to causing his relief by Eisenhower. British newspapers took their cue from Montgomery's overstatements and presently Churchill was obliged to remind the House of Commons that "The Americans have engaged thirty or forty men for every one we have engaged, and have lost sixty to eighty men to every one of ours."The primary role of British troops was in guarding the crossings of the Meuse River, the initial German objective, which freed American troops from that duty to join the battle.The British troops involved were from the XXX Corps, which included the 6th Airborne Division, the 51st (Highland) Division, the 53rd (Welsh) Infantry Division, the 29th Armoured Brigade and the 33rd Armoured Brigade. The Corps reserve was the Guards Armoured Division, the 50th (Northumbrian) Infantry Division, and the 43rd (Wessex) Infantry Division !
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American Infantry Divisions of WW1 The typical American Infantry Division was organized into what was referred to as a "square division". In 1939/1940, the US Army reduced the size of the division into what was referred to as a "triangle division". The Division was "square" in that it was built on the base number of 4. There were 4 Infantry Regiments in a Division, that were paired into Brigades. The infantry division had an artillery brigade which consisted of 3 large batteries. Also included in the unit were machine gun battalion and support units. During WW2 the typical infantry division had 3 regiments and each had 12 companies divided into 3 battalion. They had 3 artillery battalions but later a heavy battalion (155mm Howtizer) was added. This may not be the intent of your question but I tried.