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A Mystery in Finite Groups of Even Order

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GRA - Groups, representations and applications: new perspectives

Often in mathematics, we have a reason for believing that something is true, but not a proof.  I plan to discuss a result that has a proof, but no clear reason. Let x and y be elements of order two in a finite group G that are not conjugate in G.  An easy proof shows that xy has even order.  Now take an element u that lies in a normal subgroup of odd order in the centralizer  of x in G, and an analogous element v for y.  Then (xu)(yv) also has even order.   This result was obtained by simple-minded manipulation of group characters, rather than by theory or intuition about the structure of a finite group.  Suggestions for reasons are welcome.




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