Nine years ago we gave an honorary degree to Jean-Pierre Serre, at the third attempt. The first time we tried, he had to go into hospital for an operation on his knee and was unable to attend. The second time, he came the day before and gave a lecture in the School of Mathematical sciences: a beautiful, enthusiastic lecture. Sad to say, he hurt his knee doing so, and was in so much pain that he went back to Paris and was unable to wait for the ceremony. The third time, we decided that we (the Principal, Chair of Council, and a select group of mathematicians) would go to Paris instead, and the full ceremony with all the required Latin was conducted there.
The title of his lecture was On a theorem of Jordan. He didn’t tell us beforehand which theorem of Jordan he meant (there are quite a few). I
was reminded of this because last week, in a very nice colloquium talk by Mark Wildon, I learned another application of this theorem.
The theorem in question is the one asserting that, if a group G acts transitively on a set X with more than one element, then it contains an element with no fixed points. The proof is simple: it follows immediately from the fact that the average number of fixed points of elements of the group is 1: the identity fixes more than one point, so some element must fix fewer. To see this, count pairs (x,g), with x in X and g in G, such that g fixes x. Counted one way, we get the sum of fixed point numbers of elements of G; counted the other, we get the sum of the stabilisers of the elements of X, which is equal to |G| by the Orbit-Stabiliser Theorem. Dividing by |G| gives the result.
Serre began with a virtuoso translation of this proof into the languages of combinatorics, probability and analysis. Then I got my fifteen minutes of fame when he announced that it had taken 120 years for the theorem to be quantified: Arjeh Cohen and I showed in 1991 that at least a fraction 1/n of the elements of G, where n=|X|, are fixed-point-free. This is best possible: groups meeting the bound are precisely the sharply 2-transitive groups, the 1-dimensional affine groups over nearfields, which were classified by Zassenhaus in the 1930s.
He went on to describe ramifications of the theorem (and our quantification of it) in topology, number theory, etc. A paper based on this talk is published in the Bulletin of the American Mathematical Society 40 (2003), 429–440, and is well worth reading!
Jordan’s theorem is often reformulated as follows: this is how I will use it in the next result. If H is a proper subgroup of the finite group G, then not every conjugacy class in G can meet H. For if we let G act on the right cosets of H by right multiplication, then the point stabilisers are the conjugates of H, and so an element is fixed-point-free if and only if it lies in no conjugate of H, that is, its conjugacy class is disjoint from H.
Mark Wildon’s application is, as he said, nothing to set the world afire, but a nice result anyway. In a finite group G, say that two conjugacy classes C and D commute if there exist c in C and d in D such that c and d commute. (This is equivalent to the apparently stronger condition that, for all c in C, there exists d in D such that c and d commute.)
If C is a singleton class containing an element from the centre of G, then obviously it commutes with all other classes. Wildon’s observation is that the converse is true. For suppose that C commutes with all conjugacy classes, and let H be the centraliser of an element of C. Then H meets all conjugacy classes; by Jordan’s theorem, H=G.