Solvability and Order Type for Finite Groups

How much can the order type - the list of element orders (with multiplicities)—reveal about the structure of a finite group G? Can it tell us whether G is abelian, nilpotent? Can it always determine whether G is solvable? 

This last question was posed in 1987 by John G. Thompson and I answered it negatively this year. The search for a counterexample was quite a puzzle hunt! It involved turning the problem into linear algebra and solving an integer matrix equation Ax=b. This would be easy if not for the fact that the size of A was 100,000 by 10,000…