Orders, Matrices, Complex entries...

**MichaelMath** · September 23rd 2010, 10:38 AM

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

I have

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

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

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

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

So they have order 4. Correct?

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

Do I even know what I'm talking about? (just started group theory last week)

**tonio** · September 23rd 2010, 10:52 AM

Originally Posted by MichaelMath

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

I have

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

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

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

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

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 $a \rightarrow b$ 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: $<a>\cong $ because, as you proved, both these goups are cyclic of the same order (4).

Tonio

**HallsofIvy** · September 24th 2010, 05:34 AM

Originally Posted by MichaelMath

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

I have

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

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

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

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

So they have order 4. Correct?

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

Do I even know what I'm talking about? (just started group theory last week)

If, by " $a\rightarrow b$ " you mean the function that maps a into b, maps $a^2$ into $b^2$ , $a^3$ into $b^3$ , and maps $a^4= I$ into $b^4= I$ , 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..."

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