Normal subgroups, quotient groups

[kimberu]

Homework Statement

Let G be the group {$$\begin{bmatrix}{a}&{b}\\{0}&{c}\end{bmatrix}$$ | a, b, c are in $$Z_p$$ with p a prime}
Then let K = {$$\begin{bmatrix}{1}&{b}\\{0}&{1}\end{bmatrix}$$ | b in $$Z_p$$}

The map P: G --> Z*p x Z*p is defined by
P( $$\begin{bmatrix}{a}&{b}\\{0}&{c}\end{bmatrix}$$ ) = (a, c)

prove P is a group homomorphism.


I thought it would suffice to show that P(G1)P(G2)=P(G1G2) for some G1 G2 in G, and that this is true since (a1,c1)(a2,c2)=(a1a2,c1c2) -- Is this correct

Thank you so much for any help!

 

[lippyka]

</p>Homework Equations P(G1)P(G2)=P(G1G2) The Attempt at a Solution Yes, this is correct. To prove that P is a homomorphism, you must show that P(G1)P(G2)=P(G1G2) for any G1 and G2 in G. Since P is defined by P( \begin{bmatrix}{a}&{b}\\{0}&{c}\end{bmatrix} ) = (a, c), this means that P(G1)P(G2)=(a1,c1)(a2,c2)=(a1a2,c1c2). Since G1G2= \begin{bmatrix}{a1a2}&{b1+c1b2}\\{0}&{c1c2}\end{bmatrix}, it follows that P(G1G2)=(a1a2,c1c2), which is exactly what we wanted to show.