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Thread: The Size of GL_2(Z_7)

  1. #1
    Senior Member roninpro's Avatar
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    The Size of GL_2(Z_7)

    Hello everyone. I am asked to find the size of the group $\displaystyle G=GL_2(\mathbb{Z}_7)$ and would like to verify my work.

    This group acts on the vector space $\displaystyle \mathbb{Z}_7\times \mathbb{Z}_7$ with basis $\displaystyle \{(1,0), (0,1)\}$. From this, we have the orbit-stabilizer theorem: if $\displaystyle x\in \mathbb{Z}_7\times \mathbb{Z}_7$, then $\displaystyle |G|=|Gx|\cdot |G_x|$, that is, the size of $\displaystyle G$ is the size of the orbit of $\displaystyle x$ times the size of the stabilizer of $\displaystyle x$. Take $\displaystyle x=(1,0)$. If $\displaystyle A\in G$, $\displaystyle Ax$ can possibly be any nonzero vector. There are 48 of these. Therefore, $\displaystyle |Gx|=48$. Then, if $\displaystyle Ax=x$, we must have $\displaystyle A(0,1)$ equal to another vector not a multiple of $\displaystyle x$, since $\displaystyle A$ is invertible. In other words, $\displaystyle A(0,1)$ cannot be $\displaystyle (0,0), (1,0), (2,0),\ldots,(6,0)$. There are 42 of these, so $\displaystyle |G_x|=42$.

    We conclude that $\displaystyle |G|=48\cdot 42=2016$.

    Comments or suggestions?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by roninpro View Post
    Hello everyone. I am asked to find the size of the group $\displaystyle G=GL_2(\mathbb{Z}_7)$ and would like to verify my work.

    This group acts on the vector space $\displaystyle \mathbb{Z}_7\times \mathbb{Z}_7$ with basis $\displaystyle \{(1,0), (0,1)\}$. From this, we have the orbit-stabilizer theorem: if $\displaystyle x\in \mathbb{Z}_7\times \mathbb{Z}_7$, then $\displaystyle |G|=|Gx|\cdot |G_x|$, that is, the size of $\displaystyle G$ is the size of the orbit of $\displaystyle x$ times the size of the stabilizer of $\displaystyle x$. Take $\displaystyle x=(1,0)$. If $\displaystyle A\in G$, $\displaystyle Ax$ can possibly be any nonzero vector. There are 48 of these. Therefore, $\displaystyle |Gx|=48$. Then, if $\displaystyle Ax=x$, we must have $\displaystyle A(0,1)$ equal to another vector not a multiple of $\displaystyle x$, since $\displaystyle A$ is invertible. In other words, $\displaystyle A(0,1)$ cannot be $\displaystyle (0,0), (1,0), (2,0),\ldots,(6,0)$. There are 42 of these, so $\displaystyle |G_x|=42$.

    We conclude that $\displaystyle |G|=48\cdot 42=2016$.

    Comments or suggestions?
    Personally I don't see why you can't just count the possibilities based on the columns. In general $\displaystyle \displaystyle \text{GL}_n\left(\mathbb{F}_p\right)=\prod_{j=0}^{ n-1}\left(p^n-p^j\right)$. So, $\displaystyle \left|\text{GL}_2\left(\mathbb{Z}_7\right)\right|= \left(7^2-1\right)\left(7^2-7\right)=2016$.

    EDIT: And by my first sentence I really mean I don't know the theorem you're talking about but know that what follows is true haha
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