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.

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About Peter Cameron

I count all the things that need to be counted.

This observation was the concluding remark from one of the first two papers that Mark and I wrote together, Commuting conjugacy classes: An application of Hall’s Marriage Theorem to Group Theory (2009). I’m delighted to see commuting conjugacy classes getting a bit of interest. (I seem to remember getting a particularly snooty referee’s report on that paper, before we found a good home for it in the J. Group Theory!)

Thanks John. We will put a notice to this effect in our next paper on these graphs. I stumbled on the paper in J Group Theory after I wrote the above. You certainly anticipated some of the stuff we did in the first paper.

It looks as though Mark and I owe an apology of our own, to Jordan! (We failed to give him the credit for his theorem.)

One of my favourite theorems! (Especially after Serre’s talk.)