When All Subgroups Are Normal – Must the Group Be Abelian?

Group Theory Challenge: Normal Subgroups vs. Abelian Groups

Group Theory Deep Dive: A Classic Structural Puzzle

❓ The Question

Suppose G is a group where every subgroup is normal. Does this necessarily imply that G must be abelian?

💡 The Answer

No! There exist non-abelian groups where every subgroup is normal. A classic example is the quaternion group Q₈.

Details and Insight

The quaternion group Q₈ = {±1, ±i, ±j, ±k} under multiplication has:

  • 6 cyclic subgroups, all of order 4
  • 1 center {±1} of order 2

Despite being non-abelian (since ij = k ≠ ji = -k), all subgroups are normal due to Q₈’s unique structure. This makes Q₈ a Hamiltonian group – non-abelian groups where all subgroups are normal.

🔍 Why This Matters

This problem reveals that while normality often correlates with commutativity in simple cases, group structure can create surprising exceptions. It emphasizes:

  • The need to distinguish between "all subgroups normal" and "abelian"
  • How non-commutative operations can still preserve subgroup normality
  • The importance of counterexamples in algebra

📚 Further Reading

Explore these topics next:

  • Hamiltonian groups and their classification
  • Dedekind groups (alternative terminology)
  • Nilpotent vs. solvable groups

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