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Longman Dictionary of Contemporary English
major general
noun
EXAMPLES FROM CORPUS
▪ Chennault had become a major general by this time, and had his own independent military command, the Fourteenth Air Force.
The Collaborative International Dictionary
Major general

Major general \Ma"jor gen"er*al\ An officer of the army holding a rank next above that of brigadier general and next below that of lieutenant general, and who usually commands a division or a corps.

Wiktionary
major general

alt. A military officer in the armies of most nations, typically ranking below a lieutenant general and above a brigadier. n. A military officer in the armies of most nations, typically ranking below a lieutenant general and above a brigadier.

Wikipedia
Major general (United States)

In the United States Army, United States Marine Corps, and United States Air Force, major general is a two-star general-officer rank, with the pay grade of O-8. Major general ranks above brigadier general and below lieutenant general. A major general typically commands division-sized units of 10,000 to 15,000 soldiers. Major general is equivalent to the two-star rank of rear admiral in the United States Navy and United States Coast Guard and is the highest permanent rank during peacetime in the uniformed services. (The higher ranks are temporary ranks linked to specific positions, although virtually all officers who have been promoted to those ranks are approved to retire at their highest earned rank.)

Major general (Australia)

Major general (abbreviated MAJGEN) is a senior rank of the Australian Army, and was created as a direct equivalent of the British military rank of major general. It is the third-highest active rank of the Australian Army, and is considered to be equivalent to a two-star rank. A major general commands a division or the equivalent.

Major general is a higher rank than brigadier, but lower than lieutenant general. Major general is the equivalent of rear admiral in the Royal Australian Navy and air vice marshal in the Royal Australian Air Force.

The insignia for a major general is the star (or 'pip') of the Order of the Bath (despite membership of the Order no longer being awarded to Australians), above a crossed Mameluke sword and baton.

Major General (Soviet)
Major general

Major general (or major-general, short MG) is a military rank used in many countries. It is derived from the older rank of sergeant major general. The disappearance of the "sergeant" in the title explains the apparently confusing phenomenon whereby a lieutenant general outranks a major general (although a major outranks a lieutenant, a lieutenant outranks a sergeant-major.)

In the Commonwealth and United States, it is a division commander's rank subordinate to the rank of lieutenant general and senior to the rank of brigadier/ brigadier general.

In some countries, including much of Eastern Europe, major general is the lowest of the general officer ranks, with no brigadier-grade rank above colonel.

In France, major general (major général) is an appointment, normally of corps general. The major general assists a chief of staff. There is one major general for the chief of staff of the armed forces as a whole, and one for each of the four armed forces chiefs (army, navy, air force and gendarmerie).

In some small countries, such as Estonia, major general is the highest rank currently used.

In the Commonwealth, major general is equivalent to the navy rank of rear admiral, and in air forces with a separate rank structure, it is equivalent to air vice-marshal.

Usage examples of "major general".

Each army group will be composed of three divisions of approximately fifteen thousand men, each commanded by a major general.