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Proof check: find adirect product representation for Q8 grou

  1. Oct 15, 2015 #1
    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.


    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. jcsd
  3. Oct 15, 2015 #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?
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