3
$\begingroup$

Consider $G=GL_n(\mathbb{Z}/2\mathbb{Z})$. What is the smallest $n$ for which $G$ has an element of order 8? Give an example of an element of order 8.

I've thought about just considering what happens to powers of Jordan blocks since each matrix in $G$ is similar to a matrix in Jordan normal form. I've managed to convince myself that the smallest $n$ is 4, but I have no idea how to go about finding a specific element of order 8. Is there a better way than guess and check? There are far too many elements in $GL_4(\mathbb{Z}/2\mathbb{Z})$ just to guess and check.

  • 0
    @bret: yes, and in this case you can even just take x^6+x+1 = (x^9-1)/(x^3-1), and its companion matrix in GL(6,2) will have order 9. GL(5,2) has no elements of order 9. GL(3,4) and GL(2,8) both have elements of order 9 (factor$x^9-1$over these larger fields and get lower degree factors).2012-04-10

0 Answers 0