Results 1 to 4 of 4

Thread: Algebra, Problems For Fun (39)

  1. #1
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7

    Algebra, Problems For Fun (39)

    It's a well-known result that in a field $\displaystyle F,$ every finite subgroup of $\displaystyle F^{\times}$ is cyclic. A much less known but more interesting result is this:

    Let $\displaystyle D$ be a division ring, i.e. a field which is not necessarily commutative. Prove that every finite abelian subgroup $\displaystyle G$ of $\displaystyle D^{\times}$ is cyclic.


    Suggestion:
    Spoiler:
    Let $\displaystyle k$ be the center of $\displaystyle D$ and $\displaystyle C=\{c_1g_1 + \cdots + c_ng_n: \ \ n \in \mathbb{N}, c_j \in k, \ g_j \in G \}.$ Show that $\displaystyle F=\{xy^{-1}: \ x,y \in C, \ y \neq 0 \} \subseteq D$ is a field and $\displaystyle G \subseteq F^{\times}.$
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Swlabr's Avatar
    Joined
    May 2009
    Posts
    1,176
    Quote Originally Posted by NonCommAlg View Post
    Suggestion:
    Spoiler:
    ...Show that $\displaystyle F=\{xy^{-1}: \ x,y \in C, \ y \neq 0 \} \subseteq D$ is a field...
    Why have you included a $\displaystyle y^{-1}$ here? Is it known that $\displaystyle C$ is a field?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by Swlabr View Post
    Why have you included a $\displaystyle y^{-1}$ here? Is it known that $\displaystyle C$ is a field?
    $\displaystyle y^{-1} \in D,$ because $\displaystyle D$ is a division ring. the claim is that $\displaystyle C$ is a field.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Swlabr's Avatar
    Joined
    May 2009
    Posts
    1,176
    Quote Originally Posted by NonCommAlg View Post
    $\displaystyle y^{-1} \in D,$ because $\displaystyle D$ is a division ring. the claim is that $\displaystyle C$ is a field.
    ooooh-I've been getting integral domains mixed up with division rings for a long time...I thought I'd sorted that problem...ah well, this new information should certainly make this question easier!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Algebra, Problems For Fun (35)
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: Jul 21st 2009, 01:47 AM
  2. Algebra, Problems For Fun (7)
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: May 18th 2009, 05:24 AM
  3. 2 more algebra problems
    Posted in the Algebra Forum
    Replies: 1
    Last Post: Feb 5th 2009, 02:18 AM
  4. Need Help With All these Algebra Problems.
    Posted in the Algebra Forum
    Replies: 10
    Last Post: Nov 26th 2006, 05:42 PM
  5. Few more Algebra problems..
    Posted in the Algebra Forum
    Replies: 4
    Last Post: Jun 20th 2006, 05:54 AM

Search Tags


/mathhelpforum @mathhelpforum