An apology

What would life be like if I could remember all the things I ever knew?

Yesterday I was led to something I posted here twelve years ago. This was based on a talk to the London Algebra Colloquium by Mark Wildon.

He told us about results he had proved using Jordan’s theorem – this is the theorem of Jordan which gave me my fifteen minutes of fame when Jean-Pierre Serre talked about it at Queen Mary – on the existence of derangements in finite transitive permutation groups. Mark applied this to show the following, though he didn’t phrase it in these terms.

The conjugacy supercommuting graph on a group G is the graph whose vertex set is G, with an edge from x to y if there are conjugates of x and y which commute. Mark’s theorem asserted that an element of G is joined to all others if and only if it belongs to the centre of G. As a corollary, the graph is complete if and only if G is abelian.

These results form parts of theorems 4 and 5 in my paper with G. Arunkumar, Rajat Kanti Nath, and Lavanya Selvaganesh on “Super graphs on groups”, described here and now published in Graphs and Combinatorics. Our proof is the same as Mark’s.

We apologise to Mark for not attributing the result to him, and are happy to do so now.