Editing Characteristic subgroup (section)

{{Short description|Subgroup mapped to itself under every automorphism of the parent group}}
In [[mathematics]], particularly in the area of [[abstract algebra]] known as [[group theory]], a '''characteristic subgroup''' is a [[subgroup]] that is mapped to itself by every [[automorphism]] of the parent [[group (mathematics)|group]].<ref>{{cite book | last1=Dummit | first1=David S. | last2=Foote | first2=Richard M. | title=Abstract Algebra | publisher=[[John Wiley & Sons]] | year=2004 | edition=3rd | isbn=0-471-43334-9}}</ref><ref>{{cite book | last=Lang | first=Serge | authorlink=Serge Lang | title=Algebra | publisher=[[Springer Science+Business Media|Springer]] | series=[[Graduate Texts in Mathematics]] | year=2002 | isbn=0-387-95385-X}}</ref>  Because every [[Conjugacy class#Properties|conjugation map]] is an [[inner automorphism]], every characteristic subgroup is [[normal subgroup|normal]]; though the converse is not guaranteed.  Examples of characteristic subgroups include the [[commutator subgroup]] and the [[center of a group]].