Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2024 / Paper 2 / 10 / a / Solution

A group homomorphism preserves multiplication:It automatically sends the identity element to the identity and sends inverse elements to inverses. A group isomorphism is a bijective group homomorphism. For finite groups this requires equal group orders, and, once those orders agree, either injectivity or surjectivity of the group homomorphism is sufficient. Equal orders alone do not suffice.