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Assuming your revolver is a Smith & Wesson, the serial number is located on the butt of the gun. Numbers found on the frame behind the yoke (the part that holds the cylinder) are assembly numbers.
The US military never issued .32-20 revolvers. They did issue .45 caliber revolvers when they couldn't provide enough 1911 semiautos. These .45 revolvers were made by both S&W and Colt and were known as the Model of 1917 for both makers.
Most pictures of soldiers lack enough detail to ID the sidearms carried.