Cauchy numbers: job done

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 = 2p^a, 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, the list consists of the cyclic and dihedral groups of order 6 and the alternating group A₄. (Of course, in the first case, the list consists of all groups of order n.)

The paper will appear on the arXiv on Tuesday, with the same number as before.