Results 1 to 3 of 3

Thread: Orders, Matrices, Complex entries...

  1. #1
    Junior Member
    Joined
    Sep 2010
    Posts
    63

    Orders, Matrices, Complex entries...

    I'm having a little trouble with the difference between my notes and my textbook notation.

    I have

    $\displaystyle a = \[ \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right)\] $

    $\displaystyle b = \[ \left( \begin{array}{cc} i & 0 \\ 0 & -i \end{array} \right)\] $

    I have to determine the orders of $\displaystyle \left< a \right>$ and $\displaystyle \left< b \right>$, and whether $\displaystyle \left< a \right>$ and $\displaystyle \left< b \right>$ are isomorphic.

    For the first part I have $\displaystyle a^4=b^4=I_2$

    So they have order 4. Correct?

    For the isomorpic part can I just find a 2x2 matrix that shows that $\displaystyle a \rightarrow b$ isn't a homomorphism?

    Do I even know what I'm talking about? (just started group theory last week)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    3
    Quote Originally Posted by MichaelMath View Post
    I'm having a little trouble with the difference between my notes and my textbook notation.

    I have

    $\displaystyle a = \[ \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right)\] $

    $\displaystyle b = \[ \left( \begin{array}{cc} i & 0 \\ 0 & -i \end{array} \right)\] $

    I have to determine the orders of $\displaystyle \left< a \right>$ and $\displaystyle \left< b \right>$, and whether $\displaystyle \left< a \right>$ and $\displaystyle \left< b \right>$ are isomorphic.

    For the first part I have $\displaystyle a^4=b^4=I_2$

    So they have order 4. Correct?


    Yes, but also because 4 is the minimal natural power to which both matrices are I

    For the isomorpic part can I just find a 2x2 matrix that shows that $\displaystyle a \rightarrow b$ isn't a homomorphism?

    Do I even know what I'm talking about? (just started group theory last week)
    I'm afraid that the second part makes no much sense: $\displaystyle <a>\cong <b>$ because, as you proved, both these goups are cyclic of the same order (4).

    Tonio
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    19,725
    Thanks
    3008
    Quote Originally Posted by MichaelMath View Post
    I'm having a little trouble with the difference between my notes and my textbook notation.

    I have

    $\displaystyle a = \[ \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right)\] $

    $\displaystyle b = \[ \left( \begin{array}{cc} i & 0 \\ 0 & -i \end{array} \right)\] $

    I have to determine the orders of $\displaystyle \left< a \right>$ and $\displaystyle \left< b \right>$, and whether $\displaystyle \left< a \right>$ and $\displaystyle \left< b \right>$ are isomorphic.

    For the first part I have $\displaystyle a^4=b^4=I_2$

    So they have order 4. Correct?

    For the isomorpic part can I just find a 2x2 matrix that shows that $\displaystyle a \rightarrow b$ isn't a homomorphism?

    Do I even know what I'm talking about? (just started group theory last week)
    If, by "$\displaystyle a\rightarrow b$" you mean the function that maps a into b, maps $\displaystyle a^2$ into $\displaystyle b^2$, $\displaystyle a^3$ into $\displaystyle b^3$, and maps $\displaystyle a^4= I$ into $\displaystyle b^4= I$, that certainly is a homomorphism and, in fact, is the isomorphism you want.

    I don't know what you mean by "find a 2X2 matrix that shows..."
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: Jan 1st 2011, 09:54 PM
  2. Invertible Complex Matrices
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: Nov 12th 2010, 07:39 PM
  3. number of n by n orthogonal matrices with integer entries
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 18th 2010, 04:29 AM
  4. Matrices and complex numbers
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Apr 22nd 2010, 03:29 AM
  5. Matrices with integer entries
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: Dec 12th 2009, 06:54 AM

Search Tags


/mathhelpforum @mathhelpforum