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Let $p$ a prime and $n$ a positive integer, What are the outer automorphisms of the finite linear group $GL_2(\mathbb{Z}/p^{n}\mathbb{Z})$? do we know a complete list of them? Is there any thing on the literature about this?

After some search in literature, I found some types of automorphisms which are not inner, namely the following:

  • Automorphisms induced by automorphisms of the ring $R$ (This is just the identity in our case $R = \mathbb{Z}/p^{n}\mathbb{Z}$)
  • Composition of the inverse with the transpose: $T\mapsto (T^{t})^{-1}$
  • Maps of the form $T\mapsto \chi(T)T$, where $\chi$ is a group homomorphism from $GL_2(\mathbb{Z}/p^{n}\mathbb{Z})$ to the center $\{\lambda I : \lambda\in (\mathbb{Z}/p^{n}\mathbb{Z})^{\times}\}$ which satisfy $\chi(Z)\neq Z^{-1}$ for all $Z\neq I$

These types of automorphisms were called standard in literature, does this indicates the possible existence of no standard ones? Any ways I could not find a complete classification for the case I am interested on.

Rachid
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  • I'm no expert (and I'm afraid I don't know the answer to your question) but I know that there's been a lot of work on automorphisms of $\operatorname{GL}_n(R)$ for general families of rings $R$, and that typically it's harder for $n=2$ than for large $n$. I'd suggest that you wait another couple of days to see if you get an answer here, and if not then post a question on mathoverflow.net (make sure you link to this question, and add a link here to the MathOverflow question, and if you do make a post there, then it's likely to get more attention if you explain why you want to know this). – Jeremy Rickard Aug 05 '14 at 15:13
  • Thank you Jeremy for the helpful comment, Indeed it seems that the case n=2 is a difficult case, also I could not find the case when R is a ring of residual integers, Most of what I found was about integral domains and PID's. Thanks again. – Rachid Aug 06 '14 at 16:33

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