2
$\begingroup$

Suppose $F_n$ is a free group of rank $n$, and $a$ is an element of $F_n$, such that $\exists b \in F_n(b^p=a)$ and $\exists c \in F_n(c^q=a)$, where $p$ and $q$ are coprime integers. Does there always $\exists d \in F_n(d^{pq}=a)$?

It seems to be so, but I failed to find any correct proof of this statement.

Any help will be appreciated.

$\endgroup$
4
$\begingroup$

Consider the subgroup $\langle b,c \rangle$ of $F$. As a subgroup of a free group, it is itself free, but $a$ is in its centre, and the only free group with nontrivial centre is the infinite cyclic group.

So $\langle b,c \rangle = \langle g \rangle$ for some $g \in F$, and $a = g^k$ for some $k \in {\mathbb Z}$. Since $a$ is both a $p$-th power and a $q$-th power, $p$ and $q$ both divide $k$, and so $a$ is also a $(pq)$-th power.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.