# a word is zero exponent sum in free group

I read a book about free groups, it says a word is zero exponent sum, but it wasn't defined before. So what is a word which is zero exponent sum?

It means that if you write a word in the free group $F(X)$ over $X$, say, $w(x)=x_1^{e_1}\cdots x_r^{e_r}$ that $e_1+\cdots +e_r=0$. The point is, that a word in $F(X)$ belongs to the commutator group $F(X)'$ if and only if the exponent sum of each element $x_i$ in $X$ is zero. For example, the word $x^2yx^{−1}zx^{−1}y^2z^{−1}y^{−3}$ is in $F(x,y,z)'$ (e.g., the exponents of $x$ are $2,-1,-1$), but $xy^2z^{-1}x^{-1}y^{-2}$ is not.
• Hello. Where can I find a proof of this theorem you mentioned. That is, a word in $F(X)$ belongs to the commutator group $F(X)'$ if and only if the exponent sum of each element $x_i$ in $X$ is zero. Thanks. Oct 14, 2018 at 4:48
• @bfhaha See question 9 here, page $9$, with solution in the references, Lyndon, Schupp. Oct 14, 2018 at 8:15
• @Dietrich Burde Could you provide some reference for "a word in $F(X)$ belongs to the commutator group if and only if the exponent sum of each element is zero"? Thanks in advance! Feb 28, 2023 at 9:37