Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
3  
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
add comment

2 Answers

up vote 3 down vote accepted

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

share|improve this answer
    
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
add comment

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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