## Exhibit a set of representatives of the conjugacy classes of GL(3, ZZ/(2))

Exhibit a set of representatives of the conjugacy classes of $\mathsf{GL}_3(\mathbb{Z}/(2))$.

It suffices for us to give the (essentially unique) rational canonical matrix in each conjugacy class. If $A$ is a $3 \times 3$ matrix over $\mathbb{Z}/(2)$, then the minimal polynomial of $A$ has degree between 1 and 3. There are 14 such polynomials, which we list with their factorizations. (We showed which monic polynomials of degree at most 3 were irreducible in this previous exercise.)

1. $x$ (irreducible)
2. $x+1$ (irreducible)
3. $x^2 = xx$
4. $x^2+1 = (x+1)^2$
5. $x^2+x = x(x+1)$
6. $x^2+x+1$ (irreducible)
7. $x^3 = xxx$
8. $x^3+1 = (x+1)(x^2+x+1)$
9. $x^3+x = x(x^2+1)$
10. $x^3+x+1$ (irreducible)
11. $x^3+x^2 = x^2(x+1)$
12. $x^3+x^2+1$ (irreducible)
13. $x^3+x^2+x = x(x^2+x+1)$
14. $x^3+x^2+x+1 = (x+1)^3$

Note that if $x$ divides the minimal polynomial of $A$, then in particular $A$ is a zero divisor in $\mathsf{Mat}_n(F)$ and so cannot be a unit. So matrices in $\mathsf{GL}_3(\mathbb{Z}/(2))$ cannot have $x$ as a divisor of their minimal polynomial. This eliminates (1), (3), (5), (7), (9), (11), and (13) from our list of candidate minimal polynomials.

An irreducible polynomial of degree 2 is also unsuitable as a minimal polynomial, since the characteristic polynomial of $A$ (having degree 3) cannot divide any power of an irreducible polynomial of degree 2. (Either the characteristic polynomial is irreducible (bad) or has an irreducible linear factor (also bad).) This eliminates (6) from our list of candidate minimal polynomials. The six remaining candidates, once chosen as a minimal polynomial, determine the remaining invariant factors. The possible lists of invariant factors for $A$ are thus as follows.

1. $x+1$, $x+1$, $x+1$
2. $x+1$, $(x+1)^2$
3. $(x+1)(x^2+x+1)$
4. $x^3+x+1$
5. $x^3+x^2+1$
6. $(x+1)^3$

The corresponding rational canonical matrices are as follows.

1. $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$
2. $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$
3. $\begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}$
4. $\begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$
5. $\begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 1 \end{bmatrix}$
6. $\begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{bmatrix}$

These six matrices are a complete and nonredundant set of representatives of the conjugacy classes of $\mathsf{GL}_3(\mathbb{Z}/(2))$.