I'm trying to show that a group generated by elements $x,y,z$ with a given relation $xyxz^{-2}=1$ (where $1$ is the identity) is in fact a free group.
What are some usual ways of going about this kind of problem? Just hints please!
Regards
I'm trying to show that a group generated by elements $x,y,z$ with a given relation $xyxz^{-2}=1$ (where $1$ is the identity) is in fact a free group.
What are some usual ways of going about this kind of problem? Just hints please!
Regards
Are you aware of the relations of your problem to the subject of presentation of groups, in general, and in particular to the Nielsen-Schreier therorem and Nielsen transformations? This should qualify as one of the "usual ways" of going about this kind of problem at least.
You have $y=x^{-1}z^2x^{-1}$ from your relation, so $G=\langle x,z\rangle$ without any defining relation.
(This is essentially just an expansion of Boris Novikov's solution.)
There is an important technique known as Tietze transformations for manipulating presentations of groups. The two Tietze transformations are:
Add a new relation which can be derived from the existing relations. Inversely, remove a relation that can be derived from the others.
Add a new generator, together with a relation that expresses it in terms of the existing generators. Inversely, you can remove a generator if this generator only appears in one relation, and this relation defines the generator in terms of other generators.
A special case of (1) is to replace a relation with an equivalent relation. Technically, this involves two applications of (1): you must first add the new relation, and then remove the original.
Tietze's theorem is that these moves do not change the isomorphism type of a group defined by a presentation. Moreover, if two finitely presented groups are isomorphic, it is possible to get from the first presentation to the second using a finite sequence of Tietze transformations.
Now, the given group has the following presentation: $ \langle x,y,z \mid xyxz^{-2} = 1\rangle. $ The given relation is equivalent to the relation $y = x^{-1}z^2 x$, so we can replace it using two Tietze transformations of type (1): $ \langle x,y,z \mid y=x^{-1}z^2 x^{-1}\rangle. $ We can now use a Tietze transformation of type (2) to remove the generator $y$. This leaves $ \langle x,z \mid -\rangle $ which is a presentation for the free group of rank two.