# Number 4a

Has anyone else been stumped with showing 1, g, x and gx are linearly independent? It might be very trivial but me and two other classmates can’t figure it out.

Also, what significance is
there for ${x^2}={0}$?

1. Brian Cruz

I treated any product of x’s and g’s formally, so that unless the relations gave me a reason to believe that, for instance xgxgxgxgxg=ggxxggxxggx, I treated them as independent.

The relations like $x^2=0$, however, show that these are really both 0 though, so they actually are very not independent.

2. briansdumb

I think one frustrating thing about this question is that it is tempting to say “well if they aren’t linearly independent then there exists a relation not given in the description of $H_4$, so therefore they are.” 🙂
I’m hoping for a proof by contradiction, but haven’t been successful so far…

3. Brian Cruz

I think I was able to do it using the tensor algebra construction Federico talked about on here and then analyzed the generators of the ideal to show that it could not contain the necessary elements to make 1, x, g, and gx linearly dependent