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Jan 21, 2014 at 7:28 comment added Gjergji Zaimi Russ, thank you for this great answer. I will need some time to digest your latest update, but it would be wonderful if the answer turned up to be positive for all groups :-)
Jan 21, 2014 at 6:07 history edited Russ Woodroofe CC BY-SA 3.0
Greatly expanded answer to cover "most" finite groups.
Jan 15, 2014 at 0:53 history edited Russ Woodroofe CC BY-SA 3.0
Was missing the important word "normal" in the remark at near the end.
Jan 14, 2014 at 16:50 history edited Russ Woodroofe CC BY-SA 3.0
Improved argument to cover all finite solvable groups; deleted now-irrelevant discussion of supersolvable complements.
Jan 9, 2014 at 20:39 history answered Russ Woodroofe CC BY-SA 3.0