Can u help me to prove this theorem,
Two free groups are isomorphic if and only if they have the same rank.
if they have the same rank, then using the definition of a free group it's obvious that they must be isomorphic. the converse is not trivial. to prove it we'll first need a nice
lemma that extends a well-known fact about vector spaces to free modules over commutative rings with identity:
Lemma: (IBN for commutative rings) if $\displaystyle R$ is a commutative ring with identity and $\displaystyle M$ a free $\displaystyle R$-module, then every two basis of $\displaystyle M$ have the same cardinality.
Proof: let $\displaystyle m$ be a fixed maximal ideal of $\displaystyle R$ and $\displaystyle X$ any basis for $\displaystyle M.$ clearly $\displaystyle M/mM$ is a vector space over $\displaystyle R/m.$ let $\displaystyle \bar{X}=\{x+mM: \ x \in X \}.$ if $\displaystyle x_1 + mM = x_2 + mM,$
for some $\displaystyle x_1 \neq x_2$ in $\displaystyle X,$ then $\displaystyle x_1-x_2=\sum a_jx_j$ for some $\displaystyle a_j \in m$ and $\displaystyle x_j \in X,$ which is impossible since $\displaystyle X$ is basis for $\displaystyle M$ and $\displaystyle 1 \notin m,$ thus $\displaystyle |X|=|\bar{X}|.$ a similar argument
shows that $\displaystyle \bar{X}$ is a basis for $\displaystyle R/m$-vector space $\displaystyle M/mM.$ thus $\displaystyle |X|=\dim _{R/m}M/mM,$ which completes the proof because $\displaystyle \dim_{R/m}M/mM$ does not depend on $\displaystyle X. \ \ \square$
Corollary: let $\displaystyle R$ be a commutative ring with identity and $\displaystyle M$ and $\displaystyle N$ be two free $\displaystyle R$-modules with basis $\displaystyle X$ and $\displaystyle Y$ respectively. if $\displaystyle M \simeq N,$ then $\displaystyle |X|=|Y|.$
Proof: let $\displaystyle f:M \rightarrow N$ be an isomorphism. then $\displaystyle Y$ and $\displaystyle f(X)$ both are basis for $\displaystyle N,$ thus by the Lemma $\displaystyle |X|=|f(X)|=|Y|. \ \ \ \square$
now suppose $\displaystyle G_1$ and $\displaystyle G_2$ are two free groups on $\displaystyle X$ and $\displaystyle Y$ respectively. suppose $\displaystyle G_1 \simeq G_2.$ let $\displaystyle G_1'$ and $\displaystyle G_2'$ be their commutator subgroups. then clearly $\displaystyle G_1/G_1' \simeq G_2/G_2'.$
it's easy to see, from the definition of free groups, that $\displaystyle G_1/G_1'$ and $\displaystyle G_2/G_2'$ are free (obviously abelian) on $\displaystyle X$ and $\displaystyle Y$ respectively. so $\displaystyle G_1/G_1'$ and $\displaystyle G_2/G_2'$ can be considered
as two isomorphic $\displaystyle \mathbb{Z}$-modules. the above corollary now shows that $\displaystyle |X|=|Y|. \ \ \ \square$