$|GL(n,\mathbb{Z}_{p^{k}})|$

Asked Mar 03 '21 at 20:26

Active Mar 03 '21 at 20:36

Viewed 45 times

I want to calculate the order of the group $G = GL(n,\mathbb{Z}_{p^{k}})$ consist of all invertible matrices size $n\times n$ with coefficients in $\mathbb{Z}_{p^{k}}$ with $p$ a prime number, $k\in\mathbb{N}$

Where $\mathbb{Z}_{m} = \{\overline{0},\overline{1},\cdots,\overline{m-1}\}$ is the ring of integers modulo $m$

I know if we take $n=1$ we have that $|GL(1,\mathbb{Z}_{p^{k}})| = p^{k-1}(p-1)$

But if we take $n>1$, I can't calculate it.

I tried to do it using the fact that the determinant of $A\in G$ must be invertible. In other words $p \nmid det(A)$. But I'm stuck

Can someone help me?

edited Mar 03 '21 at 20:36

Alekos Robotis

26,184

asked Mar 03 '21 at 20:26

ZAF

2,618
7
13

2

First: Try to solve $k=1$, $n$ arbitrary – Hagen von Eitzen Mar 03 '21 at 20:33
Well, I'll try it!, thanks! – ZAF Mar 03 '21 at 20:35
1

When $k=1$ we have a field, and this is answered here – Derek Holt Mar 03 '21 at 20:56
Thank you so much! – ZAF Mar 03 '21 at 21:30

$|GL(n,\mathbb{Z}_{p^{k}})|$

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