# Each automorphism of the matrix algebra is inner.

Exercise I.II.IV in the book Local Representation Theory by J.L.Alperin:

Demonstrate that any automorphism of the algebra $M_n(k)$ is inner by using the fact that $M_n(k)$ has a unique simple module.

I want only hints, as this seems elementary. Thanks for your attention.

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Hint:An automorphism $\alpha$ of $M_{n}(k)$ preserves the isomorphism type of the unique simple module $V$ for $M,$ so $V$ and $V^{\alpha}$ give equivalent representations of the matrix ring. –  Geoff Robinson Aug 30 '12 at 4:56

For completeness: this is a special case of the Skolem-Noether theorem.

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Thanks for the indication. No wander I thought of the statement as elementary: I learned something before in this direction. But I forgot to refer to this theorem in pondering. Thanks again. –  awllower Aug 30 '12 at 14:31

Let me develop the comments of Geoff Robinson further into an incomplete answer.
Since $M_n(k)$ has only one irreducible representation $V$, $V^{\alpha}$ and $V$ must be the same representation. Hence the corresponding matrices are conjugate: so $\alpha$ is inner, for $M_n(k)$ is semisimple.
Point out any error which occurs please. Thanks very much.

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