Quaternion Factorization: The Hamiltonian Maximality Theorem
Mathematics is dangerous. John Carlos Baez, a Theoretical Physicist at U. C. Riverside and an excellent science communicator, tweeted about the 5/8 theorem a few days ago. Reading his tweet, I was hit by a related observation that the commutativity expectation of the quaternion group equals the number of conjugacy classes divided by the order of group. Additionally, I ‘felt’ that Hamiltonian groups must be 5/8 maximal. I felt so, because Hamiltonian groups are non-abelian Dedekind groups. In other words, despite being non-abelian, they possess a high degree of abelian-ness in that every subgroup commutes with every element of the group. Not being active in the Group theory research community, I was not sure if my observation was novel or not. What do I know? I am just a medical doctor. Nonetheless my observations and conjecture where certainly interesting to me, and I was curious to know if they are true, and more importantly if they generalized. Thus began my quest. By the end of the weekend I had named the theorem and had derived a complete original proof of it. I subsequently surmised that the theorem was almost certainly already known to be true, even though I could only find one source that alluded to it; and that source provided no accompanying proof. I learned a lot from the endeavor and drew up some future work direction for someone else. I have patients to see. I do love math but it is dangerous in that it can pull a person in very quickly without warning, hence proceed with caution.
Glossary of Terms
Group theory is a language unto itself. Much of the work of understanding it boils down to understanding its language, it. Before we proceed let’s define some key terms that we use in the theorem and proof:
Commutation: Two elements a and b are said to commute if ab=ba.
Normal: A group is said to be “normal” if every subgroup of that group is closed under conjugation. In other words, every subgroup commutes with every element of the group.
Abelian group: A group is said to be abelian if every element of the group commutes with every other element.
Dedekind group: Dedekind groups are normal groups, they may be abelian or non-abelian.
Hamiltonian group: A Hamiltonian group is a non-abelian Dedekind group
The Theorem and Proof
Here I present a theorem, the Hamiltonian Maximality Theorem, along with a proof. The theorem states that every hamiltonian group has a commutation probability of exactly 5/8. This is maximal according to the 5/8 theorem and thus demonstrates that the hamiltonian property confers the maximal abelian degree attainable for a non-abelian group. For the proof, I rely on the Dedekind-Baer theorem to represent the hamiltonian group as a product of the Quaternion group, an elementary abelian 2-group, and a periodic abelian group of odd order. And I use the centrality and conjugacy class properties of the product representation to implement a quaternion factorization that yields the result. Quaternion factorization has far-reaching implications in quantum computing.
Theorem: Hamiltonian Maximality Theorem: Every hamiltonian group has commutation probability of exactly 5/8
Proof: We present the proof via the following sequence of points:
Discussion
The 5/8 theorem as well as knowledge that the hamiltonian groups are an exact 5/8 match are not new [Koolen et al. eds. (2008); Baez et al. (2013)]. However, the latter idea seems to me to have largely eluded explicit naming and proof in the literature. We address that here. Furthermore, as noted in Koolen et al eds, P(G) = 5/8 for any G = Q8 × B where B is abelian. Our above quaternion factorization proof approach also works well for this more general case. The implications and characteristics of non-hamiltonian groups that exactly match 5/8 would indeed be interesting to explore. In particular, such groups by virtue of not being hamiltonian have some subgroups that are not normal. It is reasonable to conjecture a hierarchy of abelian degree for non-abelian groups. Clearly, being hamiltonian exceeds the minimum abelian degree required for an exact 5/8 match. A subset of non-hamiltonian groups of form Q8 × B where B is abelian are likely at the abelian degree threshold for an exact 5/8 match. Mathematical and physical insight will be gained by further investigating the parametrization and behavior around these thresholds of the diverse metrics of abelian degree, both along particular and general lines.
Acknowledgement: John Carlos Baez for drawing attention to 5/8 theorem through his blog writings.
ReferencesErdös P, Turán P. On Some Problems of a Statistical Group-Theory, IV, Acta Math. Acad. Sci. Hung. 19 (1968) 413–435Koolen J, Kwak JH, and Xu M, eds. Applications of Group Theory to Combinatorics. CRC Press. (2008) 158.Baez JC. The 5/8 Theorem. Wordpress Blog (2013)
BIO: Dr. Stephen G. Odaibo is CEO & Founder of RETINA-AI Health, Inc, and is on the Faculty of the MD Anderson Cancer Center, the #1 Cancer Center in the world. He is a Physician, Retina Specialist, Mathematician, Computer Scientist, and Full Stack AI Engineer. In 2017 he received UAB College of Arts & Sciences’ highest honor, the Distinguished Alumni Achievement Award. And in 2005 he won the Barrie Hurwitz Award for Excellence in Neurology at Duke Univ School of Medicine where he topped the class in Neurology and in Pediatrics. He is author of the books “Quantum Mechanics & The MRI Machine” and “The Form of Finite Groups: A Course on Finite Group Theory.” Dr. Odaibo Chaired the “Artificial Intelligence & Tech in Medicine Symposium” at the 2019 National Medical Association Meeting. Through RETINA-AI, he and his team are building AI solutions to address the world’s most pressing healthcare problems. He resides in Houston Texas with his family.
