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Let G be the group of all $2 \times 2 $ matrices $ \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) % \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) \] $ where $a, b, c, d$ are integers modulo $p$, $p$ a prime number, such that $ad - bc\neq 0$. $G$ forms a group relative to matrix multiplication. What is $o(G)$?

I solved a similar problem where $p=2$ by listing method. I tried solving the same problem for $p=3$ but it seems I couldn't list down all the complementary cases (i.e all the cases where $ad-bc=0$) ...but I think this is a problem which generalized the question for all primes. Is there a formula for the same i.e to calculate $o(G)$...also how to do it for $p=3$?Also, I am not aware of some concepts like rings or fields as I am learning about basic group theory... I am not quite getting it...There may some posts regarding this topic on this site...but the link provided above , although has a similar post but it is solved maybe using some concepts of rings ,fields,etc...however,I am not aware of concepts of rings,fields,column vector ,etc as I have just started reading about some basic group theory ....so that does not answer my question...

Arthur
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  • Does $o(G)$ mean the order of the group? – Philip Speegle Oct 17 '22 at 04:51
  • https://math.stackexchange.com/questions/259973/how-to-find-gl-2f-p – Philip Speegle Oct 17 '22 at 04:55
  • @PhilipSpeegle Yes..$o(G)$ denotes order of the group $G$... – Arthur Oct 17 '22 at 04:56
  • @PhilipSpeegle I don't know what $|GL_2(F_p)|$,$\text{GL}_n(\mathbb{F}_3)$ and $\text{SL}_n(\mathbb{F}_3)$ means ? As I am learning basic group theory... – Arthur Oct 17 '22 at 04:59
  • We mostly write $|G|$ for the order of $G.$ – Thomas Andrews Oct 17 '22 at 05:00
  • @ThomasAndrews Ok... – Arthur Oct 17 '22 at 05:00
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    If you know your linear algebra, $(a,b)$ can be any non-zero vector. And $(c,d)$ can be any vector which is not a scalar multiple of $(a,b).$ – Thomas Andrews Oct 17 '22 at 05:02
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    Sorry about that. $GL_2(\mathbb F_p)$ stands for the group of all $2\times 2$ matrices with entries $a,b,c,d$, all integers mod $p$ ,where $ad-bc\ne 0$. The notation $|\cdot |$ here stands for the order of the group, same as your notation $o(G)$. I think the first answer to the link is very informative. It’s exactly the argument Thomas just gave. – Philip Speegle Oct 17 '22 at 05:02
  • @PhilipSpeegle Thank you! But I don't know what is meant by column vectors in the 1st answer of the link u posted ... – Arthur Oct 17 '22 at 05:14
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    Given a matrix $\begin{bmatrix} a&b\c&d\end{bmatrix}$, we denote the columns $\begin{bmatrix} a\c\end{bmatrix}$ and $\begin{bmatrix} b\d\end{bmatrix}$ as column vectors. How familiar are you with linear algebra? – Philip Speegle Oct 17 '22 at 05:20
  • @PhilipSpeegle I have just started reading about groups.... – Arthur Oct 17 '22 at 14:05

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