These buttons are in "Record of American uniforms and Historical Buttons" page 304 under "The Grand Army Of The Republic (1866-1956) It says This organization was constituted by Union veteran of the Civil War, April 6,1866 in Decatur,Illinois. The objects of the Society were the preservation of fraternal feeling, strengthening the loyal sentiment and giving aid to needy families of veterans. The first annual encampment of "comrades" convened the following November in Indianapolis, Indiana. By 1890 nearly half a million veterans had become members. The final encampment (83rd) was held August 1949 in Indianapolis. This Grand Army of the Republic ceased to exist August 2, 1956 with the death of the last surviving member, Albert Woolson drummer boy of Company C, First Minnestoa Heavy Artillery. He was 110 years of age.
There are many variants of GAR Buttons. The Major varieties and the buttons with patented features are included. The regulation device is a monogram of Initials GAR. There are several patterns with initials not in a monogram.