Exercise 30, § 7

Let $A$ and $B$ be two groups, and let $G = A * B$ be their free product. a) Let $Z$ be an element of $A \setminus G/A$ distinct from $A$. Now prove that $Z$ contains one and only one element $z$ of the form

with $n \geq 1$, $a_i \in A - \{e\}$, $b_i \in B - \{e\}$. Now prove that every element of $Z$ is written uniquely in the form $aza'$, with $a, a'$ in $A$. b) Let $x \in G - A$. Let $f_x$ be the homomorphism from $A * A$ into $G$ whose restriction to the first (resp. second) factor of $A * A$ is the identity (resp. is the mapping $a \mapsto xax^{-1}$). Now prove that $f_x$ is injective. (Write $x$ in the form $aza'$ as above and reduce thus to the case $x = z$; use the uniqueness of the reduced decompositions in $G$.)

EXERCISES

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

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

with $n \geq 1$, $a_i \in A - \{e\}$, $b_i \in B - \{e\}$. Now prove that every element of $T$ is 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 $h_x(b) = xb x^{-1}$ if $b \in B$. Now prove that $h_x$ is injective. (The same method as in b), using the decomposition $x = atb$ of c.) In particular, we have $A \cap xBx^{-1} = \{e\}$.

Give an example where $h_x$ is not surjective.