Exercise 30, § 7

Let $A$ and $B$ be two groups and $G = A * B$ their free product.

(a) Let $Z$ be an element of $A \setminus G / A$ distinct from $A$. Show that $Z$ contains one and only one element $z$ of the form

with $n \geq 1$, $a_i \in A - \{e\}$, $b_i \in B - \{e\}$. Show that every element of $Z$ can be written uniquely in the form $aza'$ with $a, a' \in A$.

(b) Let $x \in G - A$. Let $f_x$ be the homomorphism of $A * A$ into $G$ whose restriction to the first (resp. second) factor of $A * A$ is the identity (resp. the mapping $a \mapsto xax^{-1}$). Show that $f_x$ is injective. (Write $x$ in the form $aza'$ as above and reduce it thus to the case $x = z$; use the uniqueness of reduced decompositions in $G$.)

In particular, $A \cap xAx^{-1} = \{e\}$ and the normalizer of $A$ in $G$ is $A$.

(c) Let T be an element of $A \setminus G / B$. Show that T contains one and only one element t of the form

with $n \geq 1$, $a_i \in A - \{e\}$, $b_i \in B - \{e\}$. Show that every element of T can be written uniquely in the form $atb$ with $a \in A, b \in B$.

(d) Let $x \in G$ and let $h_x$ be the unique endomorphism of G such that $h_x(a) = a$ if $a \in A$ and $h_x(b) = x b x^{-1}$ if $b \in B$. Show that $h_x$ is injective. (Same method as in (b), using the decomposition $x = atb$ of (c).) In particular, $A \cap x B x^{-1} = \{e\}$.

Give an example where $h_x$ is not surjective.