Show that any automorphism of the nxn matrices over a field k which leaves k fixed is inner. (i.e., the automorphism is "conjugation by a fixed matrix").
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I predict this is not true for representations of $S_6$ which has nontrivial outer automorphism group. – Eric Towers Jun 18 '14 at 15:09
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1@zubeyir You probably mean ring automorphism, so you should say this. Also, you have now posted several questions which are just problem statements. It would really be better if you added your partial work and progress on each problem. Readers will soon turn against you if you continue this way. – rschwieb Jun 18 '14 at 15:12
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The $k$-algebra $M_n(k)$ of $n\times n$ matrices is simple, so the result follows from the Skolem-Noether theorem. (If you're unfamiliar with either of those results, they're both reasonably easy to prove directly. The former was discussed in another question here, and the latter has a one-paragraph proof on, e.g., wikipedia.)
