In a letter to the GAP-forum Keith Dennis asks various questions about
all elements in a group being commutators of kinds. I cannot provide
any information on his properties C_n for n>1, but I can report that
in his thesis, just finished under the direction of Professor Pahlings
in Aachen, Oliver Bonten has proved that C_1 holds for 'almost all'
simple groups of Lie type. To be more precise, for each series of such
groups with fixed dimension n there is a bound for the order q of the
field involved such that C_1 holds for all groups with q bigger than
this bound. The bounds are very big and no hope to settle the
remaining cases by computer even for small n. The methods of the
proofs are from Character theory - as well as by the way were the
methods by which some years ago we verified by computer that C_1 holds
for all sporadic groups. Can one perhaps use character theory also for
verifying C_2 or C_3? I have no idea, so here is another question. If
so, we have plenty of charactertables and tools for handling them in
GAP.

Joachim Neubueser