Friday, 7 December 2007

rt.representation theory - Are there finite metabelian groups with arbitrarily many character degrees?

Can we find finite metabelian (ie with derived length 2) groups with arbitrarily many distinct degrees of irreducible complex characters?



If we cannot, can we somehow find a bound of the form $|cd(G)|leq f(dl(G))$ for some "interesting" function $f$ (linear would be very cool for instance).



The motivation is that we of course have the bound $dl(G)leq 2|cd(G)|$ for solvable groups (and conjectured to actually be $dl(G)leq |cd(G)|$), so I was wondering if a bound in the other direction also existed.

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