Final answer:
The first Italian division to be annihilated is not specifically identified in the provided information. After Italy's capitulation and Mussolini's dismissal in 1943, remaining Fascist Italian forces would have been quickly defeated or absorbed into German units during the subsequent occupation of northern Italy by Germany.
Step-by-step explanation:
The Italian division that was annihilated first during World War II was the Italian Army in North Africa.
After the Italian forces were defeated in North Africa, the Allies seized control of southern Italy, prompting Germany to invade the northern part of the country to rescue Mussolini.
Over the course of the next year and a half, Allied forces fought against German troops in Italy, slowly pushing north up the Italian peninsula.