As the new semester starts, I had to wrap up updating the revision of “graphs, groups and geometry”. I worked frantically the last few days on this as it is important to have the correction up also on the ArXiv. There was not only unexpected progress in the case of finitely generated groups (which are now all classified completely). But also towards an almost complete classification of Lie groups. For Lie groups there are now three paths to normality. A metabelian group is a group satisfying G”=1, where G’=[G,G] is the commutator group. Just as a reminder, a group is natural, if it carries a metric structure so that the algebraic structure is forced by it up to isomorphism. It turned out to be crucial what one means with “forced”. I assumed in the 2022 paper of Graphs,groups and Geometry one sided invariance mostly (like to decide about Z or D), but stated at the beginning bi-invariance. It had been Jakub Gismutallin who kindly pointed out in July of this year that this produced inconsistencies (errors) in my paper. It is a show case for Lakatos “Proofs” and “Refutations”. I had been busy in July with the “random manifold theorem” but got back to the Graphs, groups and geometry project a few days ago (and frantically focussed on fixing things). The results actually turned out to be nicer than before. This is mostly thanks to literature, which I had not been aware of in 2022, especially by Paul-Henry Leemann (a Swiss) and Mikael de la Salle (from France). Their work is actually answering almost all questions one can have about “naturality”:
Theorem 1 An arbitrary non-abelian, finitely generated group G is natural if and only if G is not generalized dicyclic.
Theorem 2 A finitely generated abelian group is natural if and only if it is finite and its Sylow 2-subgroup is elementary abelian.
In the case of Lie groups, one can say more. There are three paths to naturality:
Theorem 3: A connected Lie group is natural.
Theorem 4: An arbitrary Lie group generated by involutions is natural.
Theorem 5: A non-metabelian Lie group is natural.
All of this (and much more) can nicely be proven using rigidity theory of Leemann and de la Salle. The case of involution generated groups is illustrative and actually much more general: lets call a group involutive, if it is generated by a set S of involutions which have cardinality not larger than the continuum. We assume that every element in G can be written as a product of finitely many elements in S. An example is the locally finite symmetry group of the numbers N={1,2,3,4,….}. The key ideas for almost all results above is that one can use G to build a graph with weights so that the group itself is forced by the metric. This graph can be locally finite, (if the group is finitely generated). Its vertices or edges however also can have the cardinality of the continuum (like in the Lie group case, where the generators can be reflections). This is the originaly of the original definition: we do not care whether the metric is some sort of “usual metric”. We just want a metric.
I mentiond in the paper, where this all originally came from. We had the early years of 2000 experimented with IBL (inquiry based teaching) and invited some experts here to teach demo lessons in this way (there were IBL experts like John Neuberger who came here). There were also considerations to scale this for “service courses”. I myself once went to a conference in Texas (2005) to learn more about it. Some of these IBL folks were pretty fanatic. Education is a field, where opions can become strong. The IBL folks believe in learning a subject by “inquiry”. This can be pretty hard core: like refusing to look anything up, prove everything yourself. I myself had persued this during part of my life and found it effective sometimes, sometimes not. During college for example, the method totally crippled me because IBL is slow. If you are serious about making progress without looking up stuff or discussing stuff, you are much slower. You become an independent mathematician, but you get bad grades! I got bad grades in the first year examination at the Poly (ETH). One must know that getting into ETH is nothing. You do not have to apply. But if you fail the first examination wave (after one year), you are out. So, you better keep up with the material and learn stuff the non IBL way. (*) Any way, the “Graphs, Groups and Geometry” project was something I contemplated motivated by IBL as something one could teach and where “one can not look it up”. This is not possible today for established fields. If you google a question, the AI gods already give you the answer. This was already the case in 2005 in established fields like linear algebra or calculus. IBL fundamentally fails because staying independent and refusing to have the thinking done for you has already become almost impossible then. Today, it needs even more discipline to stay away from the machines doing things for you.
(*) I should add that I personally find it important to be exposed to a lot of knowledge. In any profession. Even if the information is at first not digested well. Exposure is important. I myelf see this (among other reasons) as one of the sources why K12 schools in the US do not well internationally. Higher education had been immune to the tendency of “dumbing down things more and more” but it started also there. I value the vast amount of exposure of mathematics, I received during college. Knowledge is fertilizer for future creativity. Yes, it temporarily dampens your creativity if you have to digest a lot of material, but the alternative, dumbing the curricula down to the lowest denominators is worse. I think that even today, where knowledge of the size of the Harvard library is available in your pocket, it is still possible to have a rich foundation of knowledge in the memory.
As a technology interested person, I of course watch the AI development too. We had experimented in 2003-2004 with our own math AI bots and it is good to see what the machines can do now. For this “graphs groups and geometry” project, I asked GPT 5.6 to investigate the ratio of “natural” to “nonnatural” groups of the order 
This reminds me of experiments we made with dogs when we were kids. One side of my grandparents always had “Cocker Spaniels”. Very nice dogs and very friendly. One of them had been a bit stupid. He would be on the balcony in the first floor and watch down to us onto the lane. I could take some bone and show it to him “Come, goody, goody!” He would bark and run around but stay in the local minimum which the distance function to the bone would provide. There was a path down of course to get the bone, but that would require to go back, take the stairs down and get back again. On the website of the American Kennel club one can see that Cocker spaniels are smart/happy/gentle. This is true but not always. The earlier dog of my grand parents was a cocker spaniel which was smart. The second one was less. He was also a bit too friendly to almost everybody. My grand father Sigbert Bader liked the first dog much more, because that one was very kind to folks it knew and not that nice to folks it did not know. But the second one was also a bit more limited, as the bone experiment showed. The picture to the right shows my grandfather Sigbert Bader with his first dog (the smart one). Anyway, I can imagine that in the near future, we human s are the pets of some AI community and that they make experiments with Humans to get entertained, like we did with dogs as kids. I think this might already happen now. Half of the population of the world seems to use AI already and I can imagine that quite a lot of stupid things are asked which one can laugh about.
Many puzzles we have studied in the God number project this spring are groups generated by involutions. Involutions were seeded in my deeply also in geometry. My geometry teacher for two courses on geometry (ordinary geometry in the first semester and differential geometry in the 5th) was Max Jeger. He had been a high school teacher also and great pedagogical talent. His pet field were “reflection groups” (Spiegelungs geometrie). It is a lost art of course now but was fancy in the 1950ies and 1960ies. Knowing that “Reflections” are cool, was important for this entire “Graphs, Groups and Geometry” project. It had started with the observation that dihedralize or Coxeterize the integers 
Theorem 4.. Any involutive group is natural.
For example, any or the orthogonal groups O(n) or Euclidean groups 
Theorem 5. Any non-metabelian finitely generated group is natural.
It is illustrative to see this work out quite nicely in Euclidean or Poincare groups. The Lorentz group O(1,3) as well as the Poincare group are involution generated so that they are natural. But they are also non-metabelian so that this is an other way to see that they are natural. The connected component of the identity of the Lorentz group is SO^+(1,3) which is not involution generated but connected and metabelian. There are still two ways to see that it is natural. The connected component of the identity in the Poincare group is also connected and non-metabelian.
An example of a Lie groups that are not involution generated, not connected and which is metabelian is the double line 
There are also handy applications for finite groups. One can immediatly seen that groups like the symmetry group 
In my 2022 version from May 27 2022 of “Graphs Groups and Geometry”, I had mentioned Larry Nichols who lived at Arlington and who had invented a version of the 2x2x2 Pocket cube in 1970. This was before Rubik as Rubik’s cube is from 1974. I learned that Nichols passed away later in 2022. Nichols lived from 1959 to 2022 in Arlington (MA), he had worked at the Moleculon Research Corporation as a Chief scientist. He is an interesting figure for me also as he lived here in Arlington. He was a chemist and got a PhD from Harvard University. I have to research a bit more about the Moleculon Research Corporation. Wikipedia knows that had been founded by Arthur Obermayer (1931-2016) who was a MIT chemist and died in Dedham Massachusetts. The website of Obermayer mentions that Moleculon was sold to an Australian corporation in 1984.
While writing this blog entry, I looked up a bit more about the Moleculon research coorporation and Nichols. The Moleculon Research corporation had been located at 139 Main street in Kendall square (across from MIT). Here is what came out “A federal Atomic Energy Commission directory already gives Moleculon at 139 Main Street in 1970; a 1976 American Chemical Society publication gives both Obermayer and Nichols at the same address; and congressional records still list Obermayer and Moleculon there in 1984.” About Nichols. Middlesex land records show Larry D. Nichols and Karen Nichols acquiring property on Gray Street on July 19, 1972, and Arlington’s municipal assessor records subsequently identify 134 Gray Street — Nichols Larry D & Karen. About the patent (from a court record) Nichols’s U.S. patent was filed March 4, 1970, but was not granted until April 11, 1972, as U.S. Patent 3,655,201, Pattern Forming Puzzle and Method with Pieces Rotatable in Groups. Moleculon was the assignee. And By the late 1980s, Moleculon Inc. was at 230 Albany Street, Cambridge, rather than 139 Main Street.
[Digging on August 24, 2026: gave an interesting twist: Wikipedia currently says that Larry D. Nichols died in July 2022 and cites an obituary. But when one follows the obituary, it is plainly a different man. That Larry D. Nichols was born December 27, 1939 in Pennsylvania, lived for decades in Lorain, Ohio, worked for Ford Motor Company, was married to Mary, served in the Marines, and had an entirely different family history. The cube inventor, by contrast, was from Xenia, Ohio, graduated from DePauw in the class of 1958, earned a doctorate at Harvard, worked for Moleculon, and lived in Arlington with his wife Karen. DePauw’s own alumni directory still associates Larry Nichols ’58 with Arlington. There is another giveaway: a 1984 UPI report describes the inventor as 47 years old, which would put his birth around 1936–37, not December 1939. So somebody appears to have attached an unrelated Ohio obituary to the inventor’s Wikipedia article, and that error has propagated. I have not yet found a reliable death notice for the Arlington inventor.]
So, there is a “Larry Nichols Mystery” (*a bit of a detective story). I will try to find out more about Larry Nichols.
Apropos Lorentz : Here are some pictures where both Lorentz and Einstein appear. (wikipedia source).
