# Mutliplication table of quotient groups

1. Nov 23, 2009

### gotmilk04

1. The problem statement, all variables and given/known data
Write the multiplication table of C$$_{6}$$/C$$_{3}$$
and identify it as a familiar group.

2. Relevant equations

3. The attempt at a solution
C$$_{6}$$={1,$$\omega$$,$$\omega^2$$,$$\omega^3$$,$$\omega^4$$,$$\omega^5$$}
C3={1,$$\omega$$,$$\omega^2$$}
The cosets are C3 and $$\omega^3$$C3
I just need help making the multiplication table.

2. Nov 23, 2009

### rasmhop

I'm assuming $C_n$ and $C^n$ both refer to the cyclic group of order n, since that's the impression I get from your post.

if you meant for $C_6$ to be generated by $\omega$, then you should have $C_3 = \{1,\omega^2,\omega^4\}$ because otherwise $C_3$ is not a group. Then the cosets should be $C_3$, $\omega C_3$.

What exactly are you having trouble with? As you said yourself the group $C_6/C_3$ has exactly two elements ($C_3$ and $\omega C_3$), so the following four are the possible products you need to compute and insert in the multiplication table:
$$C_3 \times C_3$$
$$C_3 \times \omega C_3$$
$$\omega C_3 \times C_3$$
$$\omega C_3 \times \omega C_3$$

Last edited: Nov 23, 2009