7
$\begingroup$

Here $G$ is a finite group(not neccessarily abelian),then there is a statement in some representation book that $\mathbb{Z}[G]$ is integral over $\mathbb{Z}$.That is, every element in $\mathbb{Z}[G]$ satisfies a monic polynomial equation with coefficients in $\mathbb{Z}$.

How to get this result?

I worked with the case $G=S_3$ and found it is indeed this case, and I know it also holds for the abelian case trivially, yet I have no idea how to get the general result.

Will someone be kind enough to give me some hints on this?Thank you very much!

  • 9
    Use the fact that $x$ is integral over $Z$ iff $Z[x]$ is finitely generated as a $Z$-module.2011-11-14
  • 0
    If you pick $f=a_0+a_1 g_1 + \cdots + a_n g_n\in \mathbb{Z}[G]$, you might be able to show that $f^N$ is an integer (ie all group elements are $e_G$) for some $N$ (I suspect that you might need $N>\vert G \vert$).2011-11-14
  • 0
    @JackManey:I tried that, yet failed.Let $G=S_3$ and $g\in G$ has order $3$, and let $f=e+g+g^2$, then $f^2=3f$ ,which is not an integer(though in this case the problem is solved).2011-11-14
  • 0
    @user10676:how to show that it is finitely generated?Does that mean to show that $x^N=a_0+a_1x+a_2x^2+...+a_{n}x^n$ for large $N$?Can you give some more hints?2011-11-14
  • 0
    I find that I can copy the proof of Hamilton-Cayley theorem in linear algebra word by word here for the proof of this question.Let $f\in \mathbb{Z}[G]$ define a $\mathbb{Z}$-linear hom from $\mathbb{Z}[G]$ to itself, thus corresponding to a matrix $A$ with entries all integers under the basis $\{g|g\in G\}$ sending $a$ in $G$ to $f*a$ where the mulitiplication $*$ is the one in the algebra $\mathbb{Z}[G]$.Then the proof of the H-C theorem applies here using the companion matrix of $A$.2011-11-14
  • 0
    You have to know that a sub-$Z$-module of a finitely generated $Z$-module is also finitely generated (the proof is not trivial but you can find it in any book of commutative algebra). However your proof is correct, and it is exactly how to prove the fact I claimed.2011-11-14
  • 0
    @user10676:Thank you very much!2011-11-14

3 Answers 3