Nonsimplicity

Nonsimplicity is the property of not being simple. In mathematics, a structure is called simple when it has no nontrivial substructures that are invariant under the governing operations. Nonsimplicity therefore indicates the presence of a proper, nontrivial substructure that obstructs simplicity. The precise meaning of “substructure” varies by context, such as normal subgroups in groups, ideals in rings, or invariant subspaces in other settings.

In group theory, a group is simple if it contains no nontrivial normal subgroups. Classes of nonsimple groups include many familiar examples: the cyclic group Z/nZ is nonsimple when n is composite because it has nontrivial subgroups that are normal in abelian groups; the symmetric group S3 is nonsimple since it has a normal subgroup A3. By contrast, many simple groups exist, such as the alternating group A5 and many others.

In ring theory, a ring is simple if it has no nontrivial two-sided ideals. The ring of integers Z is nonsimple because its ideals nZ for n > 0 are nontrivial. Simple rings include full matrix rings over a field, such as M_n(F). Rings that decompose into direct products or contain nontrivial ideals are nonsimple.

In other algebraic contexts, such as Lie algebras, a simple Lie algebra is non-abelian and has no nontrivial ideals. Structures with nontrivial ideals or invariant subspaces are regarded as nonsimple. Nonsimplicity commonly arises in classification and decomposition theorems, where objects are analyzed in terms of simpler, ideally simple, components.