# Order of certain subgroups of $\text{GL}_3(\mathbb{F}_p)$.

Let $$\mathbb{F}_p$$ be the field with $$p$$ element. How many elements do the following subgroups of $$\text{GL}_3(\mathbb{F}_p)$$ have?

1. $$G_1=\left\{ \begin{pmatrix} x & a & b \\ 0 & d & c \\ 0 & 0 & x \end{pmatrix} \right\}$$,
2. $$G_2=\left\{ \begin{pmatrix} x & a & b \\ 0 & x & 0 \\ 0 & d & c \end{pmatrix} \right\}$$,
3. $$G_3=\left\{ \begin{pmatrix} a & 0 & b \\ d & x & c \\ 0 & 0 & x \end{pmatrix} \right\}$$.

Here GL is the general linear group, the group of invertible n×n matrices.

Could someone provide me with a complete solution for one of the three subgroups so that I know how to redo it with the others?

The only constraint on the variables is that the determinant is nonzero. The determinant in this case is $$ax^2$$, so we must require $$a,x\ne 0$$ but $$b,c,d$$ can then be arbitrary. That means $$(p-1)^2$$ choices for $$a,x$$ and $$p^3$$ choices for $$b,c,d$$ so there are $$p^3(p-1)^2$$ elements in this subset.
• Thank you very much. As you have seen, it seems that the three subgroups have the same order $p^3(p-1)^2$. Am I right? – نورالدين سنانو Jan 26 at 12:58