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**Banach** Hi! I have got two questions concerning free groups:

a) If a group is freely generated by infinitely many elements, then it cannot be finitely generated (like the derived subgroup in F(a,b) for example). Is this true? I am pretty sure but cannot show it.

b) Let F(X) be the free group over X and x in X. Then we have a homomorphism $\displaystyle f: F(X) \to \mathbb{Z}$ which for a word w is just the sum of the exponents of x appearing in w. So for example $\displaystyle f(xyzx^2zx^{-1})=2$. Let $\displaystyle K:=\{w \in F(X): f(w)=0\}$. This is the kernel of f, hence a normal subgroup and it of course contains the group generated by X\{x}. But why does X\{x} generate K? For example $\displaystyle xyx^{-1}$ is in K, but how can it be generated from X\{x}?

Thankful for any hints,

Banach