2

Find the cardinality of the group $GL(2, \mathbb{Z/p^{n}Z}) $ for each prime $p$ and positive integer $n$ .

What I know : Clearly if $n=1$ then cardinality of $GL(2, \mathbb{Z/pZ}) $is number of ordered bases of a two dimensional vector space over the field $ \mathbb{Z/pZ}$ i.e. $(p^2-1)(p^2-p)$. Also a square matrix $A$ is invertible over the ring $\mathbb {Z/p^{n}Z}$ when $det(A)$ is not a multiple of $p$.Please help!

Arpit Kansal
  • 10,489
  • If you are given the first column, do you see what the condition is on the second column for the determinant not to be divisible by $p$? – Tobias Kildetoft Dec 12 '14 at 08:19
  • 1
    The kernel of the natural projection onto ${\rm GL}(2,p)$ consists of matrices of the form $I_2 + pM$, where $M$ is any $2 \times 2$ matrix over ${\mathbb Z}/p^n{\mathbb Z}$. That allows you to compute the order of the kernel and hence of the whole group. – Derek Holt Dec 12 '14 at 08:46
  • Related (but one answer there is wrong): https://math.stackexchange.com/questions/1169105 ; see also https://math.stackexchange.com/questions/341033. – Watson Feb 20 '21 at 17:00

0 Answers0