+0
http://www.nytimes.com/cfr/world/20071101faessay_v86n6_shaplenlaney.html?pagewanted=print
"Starting in 2008, the United States will also base a nuclear-powered aircraft carrier in Japan for the first time, bringing a new feature to its posture there. Joint military exercises between U.S. and Japanese forces have further improved integration and interoperability between the two forces, two features that were notoriously poor in the past."

How are these different:
"Joint military exercises between U.S. and Japanese forces"
"Joint military exercises of U.S. and Japanese forces"
"Joint military exercises by U.S. and Japanese forces"
"Joint military exercises among U.S. and Japanese forces"
Comments  
'Between' is best as as it emphasises the interaction of the two groups.

I could accept 'by' but don't like 'of''.

'Among' is wrong as only two groups are involved.

Rover
They are synonymous. Only among is questionable, since there are just 2 countries involved.
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Suppose there are four countries involved:

"Joint military exercises between U.S., Japanese, Canadian, and British forces"
"Joint military exercises among U.S., Japanese, Canadian, and British forces"

How are these two different (if at all)?
Two of us have already told you why among is wrong.
I think there is some misunderstanding, and it is my fault. Let's me try a different question:

"A joint military exercise between US, Japan, Canada, and Britain was conducted."
"A joint military exercise among US, Japan, Canada, and Britain was conducted."

Somebody somewhere suggested that "between" and "among" are not interchangeable and that it has to do with the number of participants.
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A joint military exercise among the US, Japan, Canada, and Britain was conducted. Don't forget the article.

Use among for three or more. Use between for two. That's the rule.
The problem is that many people, including native English speakers, make this very common error when speaking of several (more than two) things. You will see the error in print and hear it spoken, but it's still incorrect no matter how many times it's repeated.


between

‚Ė∂preposition& adverb
  • 1 at, into, or across the space separating (two objects, places, or points).
  • 2 in the period separating (two points in time).
  • 3 [as prep.] indicating a connection or relationship involving two or more parties.
  • 4 [as prep.] by combining the resources or actions of (two or more parties). (Concise Oxford English Dictionary )