Feit–Thompson theorem

The contrast that these results show between groups of odd and even order suggests inevitably that simple groups of odd order do not exist.

Many mathematicians have simplified parts of the original Feit–Thompson proof. However all of these improvements are in some sense local; the global structure of the argument is still the same, but some of the details of the arguments have been simplified.

All but two classes of maximal subgroups are of type I, but there may also be two extra classes of maximal subgroups, one of type II, and one of type II, III, IV, or V.