# Thread: Abstract Algebra: Completing coset Multiplication Tables

1. ## Abstract Algebra: Completing coset Multiplication Tables

onsider this group of six matrices:

Let G = <{I, A,B,C,D,K}, Matrix Multiplication>

I = $\left(\begin{array}{cc}1&0\\0&1\end{array}\right)$ A = $\left(\begin{array}{cc}0&1\\1&0\end{array}\right)$ B = $\left(\begin{array}{cc}0&1\\-1&-1\end{array}\right)$

C = $\left(\begin{array}{cc}-1&-1\\0&1\end{array}\right)$ D = $\left(\begin{array}{cc}-1&-1\\1&0\end{array}\right)$ K = $\left(\begin{array}{cc}1&0\\-1&-1\end{array}\right)$

Operation Table for this group:

_|I A B C D K
I |I A B C D K
A|A I C B K D
B|B K D A I C
C|C D K I A B
D|D C I K B A
K|K B A D C I

Define f: G -> <R*, $\bullet$> by y f(x) = det(x) for any Matrix x $/in$ G.

Question:

Complete the coset multiplication table for G/N (N being the Ker(f)) and Im(G) (a subgroup of <R*, MATH]\bullet[/tex]>

G/N:
_|_______
$\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$|
$\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$|
$\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$|

Image(G):
_|_________
$\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$|
$\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$|
$\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$|

I got {I, B, D} for the Ker{f} (If that's correct.)

Thanks. I appreciate any help in completing the tables.

2. This is what I think the G/N chart would be:

This is what I'm thinking for the G/N table:

__|N Na
N| N Na
Na| Na N

I found only 2 cosets and one is the Kernel and the other one I called Na due to a being one of the elements in it.

How does that look?