Welcome aboard! My first unit (Co. G, 143rd Infantry) also participated in Operation Dragoon, under the 36th Infantry Division.
I spent some of my own career (which ran from 1981-91 and 2000-11) in the Cavalry as well, including the 124th: the very last of the US Army's horse-mounted regiments, and which saw action in CBI during those days.
For further historical reading, you may wish to download LTC Steven Clay's US Army Order of Battle: 1919-1941, especially Volumes 1 and 4. All four volumes are available online, free of charge. Most of the elements under 45th Division already had a significant amount of history even before the war, and are included in this tome.
LTC Clay has dome some remarkably superb work resurrecting data from the interwar period from numerous archived sources. A seemingly incongruous point about the 45th Division is that, prior to WW2, their divisional shoulder patch bore a swastika, not the Thunderbird! Because of what the Nazis did with the swastika, the design changed quickly.