How can I show that the number of $2 \times 2$ matrices over the field of three elements with determinant $1$ is $24$?
Let $\mathbb{F}_3$ be the field with three elements. Let $n\geq 1$. How many elements do the following groups have?   $\text{GL}_n(\mathbb{F}_3)$ $\text{SL}_n(\mathbb{F}_3)$   Here GL is the , the group of invertible n×n matrices, and SL is the , the group of n×n matrices with determinant 1.
Why is $\mathbb{Z}[\sqrt{-n}], n\ge 3$ not a UFD? Built-in text editor in Terminal Is there a built-in text editor in the terminal on Lion?  (Like, for example, nano on Linux.) What will reduce my rendertime? What would reduce the render time? Should I get a better GPU or more RAM? How to make a file executable? How can I make my file so that I can double click on it, and it runs. It is a .sh script, but I also have files that say:  executable (application/x-executable)  in the description of what they are.  I can't run any of these from terminal, or by double clicking.  If possible, I would like a way using either the GUI or a Terminal, but not a combination of the two.  Here is a screenshot of what I see when I right click then go on properties. The file first:    And the shell script here:    NB: I accept that this is a duplicate (I was looking for it, and couldn't find it, so asked + answered it, hoping that I would find it) however, I don't think the question about .desktop files is a duplicate. I would like to show that ring of order $p^2$ is commutative.   Taking $G=(R, +)$ as group, we have two possible isomorphism classes $\mathbb Z /p^2\mathbb Z$ and $\mathbb Z/ p\mathbb Z \times \mathbb Z /p\mathbb Z$.   Since characterstic must divide the size of the group then we have two possibilities $p$ and $p^2$.   Now IU don't understand how can I reason to say that the multiplication is commutative and how can I conclude for the case when characterstic is $p$? What is the net charge of the Earth? How to use Redstone dust in Minecraft PE? I have Minecraft Pocket Edition and I have collected a lot of redstone dust. For some reason, I couldn't use it for things such as iron doors. Why? 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