For the past one week, I have been trying to learn more about automorphism groups of different groups. Very recently one of my friend asked this question to me:
- What is the automorphism group of $(\mathbb{Q}^{\ast},\times)$. In short, what is $\text{Aut}(\mathbb{Q}^{\ast})$?
I emailed couple of friends and got the answer as:
- $\text{Aut}(\mathbb{Q}^{\ast})$ is isomorphic to the automorphism group of a free abelian group of countable rank. In particular, it will contain $\text{GL}(n,\mathbb{Z})$ for all $n$.
My question would be :
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Can we realize $\text{SL}_{n}(\mathbb{Z})$ to be the automorphism group of some group?
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Are there groups which are which are "very difficult" to be realized as the automorphism group of a certain group.
So suppose someone comes and asks me: Is $S_{3}$ or $\text{GL}_{2}(\mathbb{Z})$ the automorphism group of some group, then how can I answer the question? I am particularly interested in seeing how to think for a solution.
Best Answer
This doesn't quite answer your question, but Bumagin and Wise proved every countable group is the outer automorphism group of a finitely generated group.