# Proof check: find adirect product representation for Q8 grou

1. Oct 15, 2015

### davidbenari

1. The problem statement, all variables and given/known data
Find a direct product representation for the quaternion group. Which are your options?

2. Relevant equations

3. 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.

Lets 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?

2. Oct 15, 2015

### davidbenari

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?