I was doing one of the proofs for my abstract algebra class, and we had to prove that the cardinality of the image of G, [θ(G)] is a divisor lGl. I'm trying to intuitively understand why G and it's image don't necessarily have the same cardinality. I'm thinking it's because there isn't necessarily a one to one correspondence, if there is then they have the same cardinality, but if they don't then the cardinality of the image is less than the cardinality of G, am I thinking along the right track here? This isn't a homework problem, but I came across it in my work. Any insight is appreciated.