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.

  • 3
    $\begingroup$ 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. $\endgroup$ – Geoff Robinson Aug 30 '12 at 4:56

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

  • $\begingroup$ 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. $\endgroup$ – 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.