1. The problem statement, all variables and given/known data
Note: I did not get this problem from a textbook.
Let
denote the (nonabelian) free group on the
generators
, and let
be arbitrary. My question is, does there exist a
such that
, besides
(the identity); is such an equation in the free group possible? Obviously this equation would imply they commutate, but that won't necessarily be a contradiction. It appears that I am effectively asking whether the centralizer of a given element in
is nontrivial. I can't determine the answer to this question; perhaps someone would be so kind as to guide me towards it
2. Relevant equations
3. The attempt at a solution
Note: I did not get this problem from a textbook.
Let
2. Relevant equations
3. The attempt at a solution