# Thread: Orders, Matrices, Complex entries...

1. ## 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)

2. Originally Posted by MichaelMath
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

3. Originally Posted by MichaelMath
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..."