Proof check: find adirect product representation for Q8 grou

[davidbenari]

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?

 

[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?