Proof check: find adirect product representation for Q8 grou

In summary, the conversation discusses finding a direct product representation for the quaternion group, with a focus on using normal subgroups. The possibility of forming a product with proper subgroups is explored, and the theorem of internal direct products is mentioned. There are also doubts about the applicability of the theorem and the necessity of using subgroups in forming a direct product representation.
  • #1
davidbenari
466
18

Homework Statement


Find a direct product representation for the quaternion group. Which are your options?

Homework Equations

The Attempt at a Solution


Theorem: The internal direct product of normal subgroups forms a homomorphism of the group.

https://proofwiki.org/wiki/Internal_Group_Direct_Product_of_Normal_Subgroups

The quaternion group as 6 normal subgroups, 4 of which are proper.

Let's suppose I only choose proper subgroups. The orders of these are 2,4,4,4.

Then I can form the product (grouporder2)x(anyothergrouporder4) and assure myself that it is a isomorphism.

Therefore, I've created a direct product representation.

Is this correct?
 
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  • #2
Note that the intersection of my subgroups is has 2 elements, therefore I think the theorem doesn't apply.

Also, I'm having doubts as to why any direct product representation of a group has to be made via subgroups. How can I prove this?
 

1. What is a direct product representation?

A direct product representation is a way of expressing a group as the product of other groups. This means that every element in the original group can be written as a combination of elements from the other groups.

2. What is the Q8 group?

The Q8 group, also known as the quaternion group, is a non-commutative group with 8 elements. It is denoted by Q8 = {1, -1, i, -i, j, -j, k, -k} and has the following multiplication table:

• 1 -1 i -i j -j k -k
1 1 -1 i -i j -j k -k
-1 -1 1 -i i -j j -k k
i i -i -1 1 k -k -j j
-i -i i 1 -1 -k k j -j
j j -j -k k -1 1 i -i
-j -j j k -k 1 -1 -i i
k k -k j -j -i i -1 1
-k -k k -j j i -i 1 -1

3. How do you find a direct product representation for a group?

To find a direct product representation for a group, you need to identify two subgroups of the group that are disjoint (meaning they have no common elements), have the same order, and their product results in the entire group. The direct product of these two subgroups will then be a representation of the original group.

4. What are the subgroups of the Q8 group?

The subgroups of the Q8 group are {1, -1}, {1, -1, i, -i}, {1, -1, j, -j}, and {1, -1, k, -k}. These subgroups are all disjoint, have the same order (2

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