This Verma module has an irreducible quotient by its largest nontrivial submodule. More generally, any one-dimensional representation is irreducible by virtue of having no proper nontrivial subspaces. Therefore any space with a nontrivial homotopy group cannot be contractible. The integers are the only nontrivial totally ordered abelian group whose positive elements are well-ordered. A set can be regarded as an algebra having no nontrivial operations or defining equations. The smallest example with a nontrivial subgroup of false witnesses is . Every nontrivial monotone polygonal chain is open. The applied methods usually refer to nontrivial mathematical techniques or approaches. For example, let formula 32 be a nontrivial linear representation. Note that this algorithm may fail to find a nontrivial factor even when is composite. This was an early example of a nontrivial duality of categories. The only nontrivial equation is the one for formula 35. In fact, the free product of any two nontrivial groups is infinite. The only nontrivial square Fibonacci number is 144. No one had realized that set theory had any nontrivial content.