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.

About Peter Cameron

I count all the things that need to be counted.
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4 Responses to An apology

  1. John Britnell says:

    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.

  2. jrbritnell says:

    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.)

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