- Let F(A) and F'(A) both satisfy the universal property of a free group on a set A. Show that there is a unique isomorphism from F(A) to F'(A) that is compatible with the canonical maps A -> F(A) and A -> F'(A). Hint: consider the universal property for all possible combinations, taking both F(A) and F'(A) to be the free group and the supplementary group.
- Can you see how a similar argument could be used to show quite generally that objects that satisfy universal properties are unique up to isomorphism?
- DF 5.4 #2,
- DF 5.4 #7,
- DF 5.5 #23-a