I am currently taking introduction to algebra and I am struggling to find rapidly isomorphism between groups. I know my question is a bit broad in a sense that there is no singular answers but I find it cumbersome to find (1) a potential isomorphism and (2) to verify that this potential isomorphism is indeed an isomorphism. For example, one kind question that I struggle with ; Show two isomorphisms between U(20) and U(16). What I would do in practice is list the order of each elements of these groups and construct an isomorphism such that elements of same order map to each other but in an exam, I wouldn't have the time to do so. To conclude, is there some kind of aptitude that I should master to spot isomorphisms quickly?
Thank you very much