Free groups and isomorphism

Question

Is it true that if $F_a(X)$ is the free group with free basis $X$, and the same for $F_a(Y)$, and $X,Y$ have the same cardinality, then there is an isomorphism between $F_a(X)$ and $F_a(Y)$?

Answer 1

Hint: You know that there is a bijection $X \to Y$; how might you extend this to a group isomorphism?

Answer 2

Yes, this follows immediately from the universal property of a free group. If $\phi: X \to Y$ is a bijection, then composing it with inclusion $Y \subset F(Y)$ we get function $X \to F(Y)$ which uniquely determines a homomorphism $F(X) \to F(Y)$ that maps bijectively $X \subset F(X)$ to $Y \subset F(Y)$. Similarly, considering $\phi^{-1}: Y \to X$, we get a unique homomorphism $F(Y) \to F(X)$ that maps $Y \subset F(Y)$ to $X \subset F(X)$. Now the composition of these two is identity.