Proof check: find adirect product representation for Q8 grou

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SUMMARY

The discussion focuses on finding a direct product representation for the quaternion group, which consists of six normal subgroups, four of which are proper. The participant proposes using the internal direct product theorem, asserting that a product of a group of order 2 and any group of order 4 can yield an isomorphism. However, they express uncertainty regarding the application of the theorem due to the intersection of subgroups containing two elements and question the necessity of using subgroups for direct product representations.

PREREQUISITES
  • Understanding of group theory concepts, specifically the quaternion group.
  • Familiarity with normal subgroups and their properties.
  • Knowledge of the internal direct product theorem in group theory.
  • Basic skills in proving group isomorphisms and subgroup intersections.
NEXT STEPS
  • Study the properties of the quaternion group and its normal subgroups.
  • Learn about the internal direct product theorem and its applications in group theory.
  • Research methods for proving group isomorphisms, particularly in the context of direct products.
  • Explore the implications of subgroup intersections on direct product representations.
USEFUL FOR

Students of abstract algebra, mathematicians specializing in group theory, and anyone interested in the structural properties of the quaternion group and direct product representations.

davidbenari
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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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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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