# 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

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.