Editing Characteristic subgroup (section)

=== Finite example ===
Consider the group {{math|''G'' {{=}} S{{sub|3}} × <math>\mathbb{Z}_2</math>}} (the group of order 12 that is the direct product of the [[symmetric group]] of order 6 and a [[cyclic group]] of order 2). The center of {{math|''G''}} is isomorphic to its second factor <math>\mathbb{Z}_2</math>. Note that the first factor, {{math|S{{sub|3}}}}, contains subgroups isomorphic to <math>\mathbb{Z}_2</math>, for instance {{math|{e, (12)} }}; let <math>f: \mathbb{Z}_2<\rarr \text{S}_3</math> be the morphism mapping <math>\mathbb{Z}_2</math> onto the indicated subgroup. Then the composition of the projection of {{math|''G''}} onto its second factor <math>\mathbb{Z}_2</math>, followed by {{math|''f''}}, followed by the inclusion of {{math|S{{sub|3}}}} into {{math|''G''}} as its first factor, provides an endomorphism of {{math|''G''}} under which the image of the center, <math>\mathbb{Z}_2</math>, is not contained in the center, so here the center is not a fully characteristic subgroup of {{math|''G''}}.