Exercise 26, § 7

Let X be a set and g an element of the free group $F(X)$. Let $g = \prod_{\alpha=1}^n x_\alpha^{m(\alpha)}$ be the canonical decomposition of g as a product of powers of elements of X (cf. no. 5, Proposition 7). The integer $l(g) = \sum_\alpha |m(\alpha)|$ is called the length of g. (a) g is called cyclically reduced if either $x_1 \neq x_n$ or $x_1 = x_n$ and $m(1)m(n) > 0$. Show that every conjugacy class of $F(X)$ contains one and only one cyclically reduced element; it is the element of minimum length of the class in question. (b) An element $x \in F(X)$ is called primitive if there does not exist an integer $n \geq 2$ and an element $g \in F(X)$ such that $x = g^n$. Show that every element $z \neq e$ of $F(X)$ can be written uniquely in the form $x^n$ with $n \geq 1$ and x primitive. (Reduce it to the case where z is cyclically reduced and observe that, if $z = x^n$, the element x is also cyclically reduced.) Show that the centralizer of z is the same as that of x; it is the cyclic subgroup generated by x.