A request

In 1956, Helmut Wielandt proved that a primitive permutation group whose degree is twice a prime p is doubly transitive, unless p has the form 2a2+2a+1, in which case the group has rank 3, and its subdegrees are a(2a+1) and (a+1)(2a+1). The only examples known to Wielandt at the time were the symmetric and alternating groups of degree 5 acting on the 2-element subsets of {1,…,5}. Now, using the Classification of Finite Simple Groups, we know that there are no others.

Wielandt’s proof falls into two parts. First it is shown that such a group has rank 3 and the permutation character is the sum of irreducible characters of degrees 1, p−1 and p. The second part of the proof starts from this decomposition and finds the form for p and the subdegrees.

As noted above, the theorem Wielandt published is now superseded. However, the second part of his proof gives a combinatorial result, using neither the primality of p nor the existence of a primitive permutation group. It is a theorem about strongly regular graphs: these are regular graphs (so the all-1 vector is an eigenvector of the adjacency matrix whose eigenvalue is the valency), with the property that on the space orthogonal to the all-1 vector the adjacency matrix has just two eigenvalues. The result is stated as Theorem 2.20 in my book with Jack van Lint. It says:

Theorem Suppose that a strongly regular graph on 2n vertices has the property that the restriction of the adjacency matrix to the space orthogonal to the all-1 vector has eigenvalue multiplicities n−1 and n. Then, up to complementation, either

  • the graph is the disjoint union of n complete graphs of size 2; or
  • the parameters are those given by Wielandt, for some integer a.

There are a number of examples of the second case beyond those of degree 10 (the line graph of K5 and its complement the Petersen graph; for example, the strongly regular graphs on 26 vertices with valency 10.

Now my question is this:

Question Who first noticed that Wielandt’s argument gave this more general result?

The reference given above is the earliest one I know, but it seems unlikely that we were the first to see this.

About Peter Cameron

I count all the things that need to be counted.
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