free object isomorphisms

Question

In group category if $F_1$ be a free object on $X_1$ and $F_2$ is free object on $X_2$ and $F_1$ isomorphic to $F_2$ prove that |$X_1$|=|$X_2$|

whats the relationship between isomorphisms of free objects and cardinality of $X_i$ ?

Answer 1

Consider the abelization $G^{ab}=G/{[G,G]}$, where $[G,G]$ is the commutator-subgroup of $G$.

If $F_1\cong F_2$, we get $$\mathbb{Z}^{|X_1|}\cong F_1^{ab}\cong F_2^{ab}\cong \mathbb{Z}^{|X_2|}.$$ Tensoring with $\mathbb{Q}$ over $\mathbb{Z}$, one gets $$\mathbb{Q}^{|X_1|}\cong \mathbb{Z}^{|X_1|}\otimes\mathbb{Q}\cong\mathbb{Z}^{|X_2|}\otimes\mathbb{Q}\cong\mathbb{Q}^{|X_2|}.$$ This means we have an isomorphism $\mathbb{Q}^{|X_1|}\cong\mathbb{Q}^{|X_2|}$ of $\mathbb{Q}$-vectorspaces and since the dimension of vectorspaces is well-defined even in infinity dimensions (dimension theorem) we get $|X_1|=|X_2|$.