find an element of order p in GL2(Zp)

Question

I am kind of stuck on one of my homework questions, which asks:

Let $\mathbb{Z}_p$ denote the integers modulo $p$. Find an element of order $p$ in $\mathbb{GL}_2(\mathbb{Z}_p)$. Can you also find an element of order $2p?$

I understand the concept of general linear group and modulo class but I am in need of some hints about finding these kinds of element.

Thanks a lot.

I assume $p$ is meant to be prime and the question talks about $GL_2(Zp)$ not $GL(Zp)$. — Mark Fischler, Apr 02 '18 at 23:17
See also https://math.stackexchange.com/questions/79047/on-the-order-of-elements-of-gl2-q — lhf, Apr 02 '18 at 23:35
To find an element of order $2p$, you'll have to assume $p \ne 2$. Otherwise, $GL_2(\mathbb{Z}_2) \simeq S_3$ which has no elements of order 4. — Daniel Schepler, Apr 02 '18 at 23:39

lhf · Accepted Answer · 2018-04-03T01:15:54.580

5

Hint:

Consider matrices of the form $\pmatrix{ 1 & b \\ 0 & 1}$.
Consider matrices of the form $\pmatrix{ -1 & \hphantom{-}b \\ 0 & -1}$.

Handle $p=2$ separately.

edited Apr 03 '18 at 01:15

answered Apr 02 '18 at 23:29

lhf

216,483

So can you find an element of order $2p=6$ when $p=3$? – Mark Fischler Apr 02 '18 at 23:34
Thanks for the hint. So I can take b to be 1 in order to have 1p = 0 mod p but how can I get another element to have order 2p? I am thinking about 1/2 but clear it's illegal – Aaron Li Apr 02 '18 at 23:42
1

Well, $1/2\equiv\frac{p+1}2\pmod p$, so $1/2$ isn’t illegal, it’s just no help. – Lubin Apr 02 '18 at 23:47

find an element of order p in GL2(Zp)

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