Page "Minimal counterexample" Paragraph 1

If the form of the contradiction is that we can derive a further counterexample D, and that D is smaller than C in the sense of the working hypothesis of minimality, then this technique is traditionally called infinite descent.

There may however be more complicated ways to argue.

For example, the minimal counterexample method has been much used in the classification of finite simple groups.

The Feit – Thompson theorem, that infinite simple groups that are not cyclic groups have even order, was based on the hypothesis of some, and therefore some minimal, simple group G of odd order.

Every proper subgroup of G can be assumed a solvable group, meaning that much theory of such subgroups could be applied.