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Regiment or battalion: 600 to 1000 troops. Usually led by a Colonel.
Brigade: typically three battalions (or regiments in the USA, USSR - different names, same thing). 2000 to 5000 troops. Led by a Brigadier General.
Division: Usually three brigades, anywhere between 10,000 to almost 30,000 men. Led by a Major General, a division is the smallest unit that is normally expected to be a complete, organic all-arms force: i.e., a division has its own infantry, tanks, artillery, engineers, transport, and everything else required to make up an entire fighting force. The classic "triangular" formulation (as used by Napoleon, who was one of the first to make use of this concept, and still used by the majority of armies today is to have three brigades, plus a regiment each of artillery, cavalry, and engineers, which are more-or-less permanantly attached to the division.
Some armies, notably the German Army in the second half of WW2, have only two brigades to the division. This is mainly designed, for any given number of troops, to make the overall force look larger. "Look!" Hitler could scream at his generals, "We have 7 divisions investing Minsk! Why are we not seeing more progress!" Alternatively, some armies (notably the US Army during WW1) have "square" divisions: 4 brigades per division. This makes each division very powerful. On the other hand, you don't get to have so many of them. The classic division, however, is the "triangular" division: 3 brigades. This is the usual practice. Generally, in modern armies (like the US Army in the 1991 Gulf War), you have 1 infantry + 2 cavalry brigades for an "armour heavy" division, or else 1 cavalry + 2 infantry for a "infantry heavy" division. Commanders tend to swap brigades over between different divisions depending on the type of fighting expected - more infantry for house to house stuff or congested terrain, more tanks for battles of movement in open country.
One of the great advantages of the triangular division is that it is so easy to have a balanced force structure: typically "two up and one down" - i.e., two brigades in the front line, the third in reserve to be commited as needed. (But sometimes the other way about.) Napoleon was no fool.
Corps: Two or more divisions. Commanded by a lieutenant general.
Army: Two or more corps. Commanded by a full general.
There will be a short test after class.