Originally Posted by

**kimberu** Let G be the group {$\displaystyle

\begin{bmatrix}{a}&{b}\\{0}&{c}\end{bmatrix}

$ | a, b, c are in $\displaystyle Z_p$ with p a prime}

Then let K = {$\displaystyle

\begin{bmatrix}{1}&{b}\\{0}&{1}\end{bmatrix}

$ | b in $\displaystyle Z_p$}

The map P: G --> $\displaystyle Z_p*$ x $\displaystyle Z_p*$ is defined by

P( $\displaystyle

\begin{bmatrix}{a}&{b}\\{0}&{c}\end{bmatrix}

$ ) = (a, c)

(A). Prove the quotient group G/K is isomorphic to $\displaystyle Z_p*$ x $\displaystyle Z_p*$

(B). Find the orders |g|, |K|, and |G/K|

my thoughts -

(A) I've already shown that K is normal in G, and I know this would be true if K were the kernel of P, but I don't know how to show that.

(B), I'm not sure -- is it |G|= 3|$\displaystyle Z_p$|, |K|=|$\displaystyle Z_p$|, and |G/K| = 2|$\displaystyle Z_p$|?

Thank you so much for any help!