Cauchy numbers: job done
Peter Cameron's Blog
by Peter Cameron
1M ago
After recruiting Scott Harper to the team, we have finished the job of determining all the Cauchy numbers (these are the positive integers n for which there exists a finite list F of finite groups so that a finite group has order divisible by n if and only if some member of F is embeddable in it. The answer is: n is a Cauchy number if and only if one of the following holds: n is a prime power; n = 6; n = 2pa, where p is a Fermat prime greater than 3 and a is a positive integer. In the second and third cases, we can tell you the list F: for example, for n = 6, th ..read more
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Anatoly Vershik
Peter Cameron's Blog
by Peter Cameron
1M ago
Anatoly Vershik died the day before yesterday. As I have told here, he was the person who told me about the Urysohn space. I had given a talk at the ECM in Barcelona on the countable random graph, and after it he approached me and asked “Do you know about the Urysohn space?” We wrote a paper on it, extending some of my results on the random graph. Indeed, the Urysohn space has many different Abelian group structures. He also made my two trips to St Petersburg possible, by having birthdays in 2004 and 2014 (actually his birthdays were in late December the previous year). I learned so much at th ..read more
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Richard Parker
Peter Cameron's Blog
by Peter Cameron
1M ago
Richard Parker died last month. Now only two of the authors of the ATLAS of finite groups remain, the two Robs. I knew Richard, but perhaps not well enough to write anything appropriate as a tribute. But I recommend you take a look at Rob Wilson’s account on his blog ..read more
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More on Cauchy numbers
Peter Cameron's Blog
by Peter Cameron
2M ago
Following on from the earlier post, the new version of the paper has just gone on the arXiv: 2311.15652 (version 2). If we say that n is a Cauchy number if there is a finite set F of finite groups, all with orders divisible by n, such that every group with order divisible by n must contain a group in F as a subgroup, then our result is as follows: Let n be the product of two distinct primes q and r. Then n is a Cauchy number if and only if one of the primes is 2 and the other is a Fermat prime. This means that, in all other cases, there are infinitely many groups which are minimal with respect ..read more
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Cauchy’s theorem for the prime 6
Peter Cameron's Blog
by Peter Cameron
2M ago
Before you think I have gone totally crackers: Cauchy’s theorem says that a finite group whose order is divisible by a prime number p contains a subgroup which is cyclic of order p. My co-authors and I have proved some similar results, of which the one referred to in the title is the following: A finite grup whose order is divisible by 6 contains a subgroup which is either cyclic of order 6, dihedral of order 6, or isomorphic to the alternating group of degree 4 (with order 12). When the more general theorem is proved and the paper written, I hope to elaborate on this. But my question for now ..read more
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Simon Norton lecture
Peter Cameron's Blog
by Peter Cameron
2M ago
I have been honoured by an invitation to give the inaugural Simon Norton lecture at the London Institute for Mathematical Sciences on 12 February. The webpage is here. Of course there are other people who knew Simon better than I did. Nevertheless, I think I have something to say. It was a paper of mine, with Jean-Marie Goethals and Jaap Seidel, which (as far as I know) introduced the term Norton algebra for a commutative but non-associative algebra of the general type which Bob Griess later used to construct the Monster. Our reference for this is to a personal communication from J. H. Conway ..read more
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New Year
Peter Cameron's Blog
by Peter Cameron
3M ago
The youth could not help breaking a rule of courtesy towards this heavily burdened and yet, as he felt, noble man by asking: “But tell me, I beseech you, why do you carry on such wars on your star? Who is to blame for them? Are you yourself in part responsible?” The King seemed angered at this audacity and for a time stared at the messenger. But he could not continue to meet with his dark gaze the bright and guileless eyes of the stranger. “You are a child,” said the King, “and there are things you cannot understand. War is no one’s fault, it occurs of itself, like storm and lightning, and all ..read more
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Programming and typesetting
Peter Cameron's Blog
by Peter Cameron
3M ago
Is computer typesetting a kind of programming? One of the pioneers, Donald Knuth, clearly thought so. In The TeXBook, he gives TeX code for computing and typesetting the first thirty primes; apart from anything else, this demonstrates that TeX has the capacity to act as an all-purpose program. But there is a rather significant difference. A programmer requires the program to deliver the right answer: exactly, for a discrete mathematician, and to with a specified approximation, for a continuous mathematician. Computer typesetting doesn’t deliver an answer as such. Knuth realises this when The T ..read more
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Dominic Welsh
Peter Cameron's Blog
by Peter Cameron
4M ago
Dominic Welsh died late last week. Dominic and I were tutors at Merton College, Oxford for nearly eleven years. I was the pure maths tutor and he was the applied maths tutor. But there was no other Oxford college where the mathematical interests of the pure and applied maths tutors were closer. College lunches are an opportunity to talk to colleagues in other areas; Dominic and I broke this convention to some extent by discussing our research, though he was in no way an antisocial mathematician! He had a very great impact on the subject. Much of this came from his provocative conjectures, but ..read more
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Covers of groups
Peter Cameron's Blog
by Peter Cameron
4M ago
Here is the second of these discussions of elementary problems on finite groups with unexpected ramifications. This topic was suggested to me by Hamid Reza Dorbidi form Jiroft, Iran. Let F be a finite set of finite groups. A finite group G is a cover of F if every group in F is isomorphic to a subgroup of G; it is minimal if no proper subgroup of G is a cover, and minimum if no group of smaller order than G is a cover. Covers certainly exist; for example, the direct product of the groups in F is a cover of F. But we would like more economical covers, that is, minimal or (especially) minimum co ..read more
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