# Mutliplication table of quotient groups

• gotmilk04
In summary, the conversation was about writing the multiplication table for the cyclic group C_6/C_3 and identifying it as a familiar group. The group was defined as C_6 = {1, ω, ω^2, ω^3, ω^4, ω^5} and C_3 = {1, ω, ω^2}, with the cosets being C_3 and ωC_3. The task was to compute the products of these cosets and fill in the multiplication table.
gotmilk04

## Homework Statement

Write the multiplication table of C$$_{6}$$/C$$_{3}$$
and identify it as a familiar group.

## 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.

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:

## 1. What is a multiplication table of quotient groups?

A multiplication table of quotient groups is a table that shows the results of multiplying elements from different quotient groups. It is used to understand the structure and relationships between the elements of quotient groups.

## 2. How is a multiplication table of quotient groups constructed?

A multiplication table of quotient groups is constructed by first selecting two quotient groups and their respective elements. Then, the elements are multiplied according to the operation defined in the quotient group and the resulting element is placed in the corresponding row and column of the table.

## 3. What is the purpose of a multiplication table of quotient groups?

The purpose of a multiplication table of quotient groups is to provide a visual representation of the relationships between elements in different quotient groups. It can also be used to perform calculations and determine the structure of the quotient groups.

## 4. Can a multiplication table of quotient groups be used to compare different quotient groups?

Yes, a multiplication table of quotient groups can be used to compare different quotient groups. By comparing the multiplication tables, we can observe similarities and differences between the structures of the quotient groups.

## 5. Are there any limitations to using a multiplication table of quotient groups?

One limitation of using a multiplication table of quotient groups is that it can become quite large and complex for larger quotient groups. Additionally, it may not provide a complete understanding of the relationships between elements in the quotient groups, and other mathematical tools may be needed to fully analyze them.

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