Editing Center (group theory) (section)

==As a subgroup==
The center of ''G'' is always a [[subgroup (mathematics)|subgroup]] of {{math|''G''}}.  In particular:
# {{math|Z(''G'')}} contains the [[identity element]] of {{math|''G''}}, because it commutes with every element of {{math|''g''}}, by definition: {{math|1=''eg'' = ''g'' = ''ge''}}, where {{math|''e''}} is the identity;
# If {{math|''x''}} and {{math|''y''}} are in {{math|Z(''G'')}}, then so is {{math|''xy''}}, by associativity: {{math|1=(''xy'')''g'' = ''x''(''yg'') = ''x''(''gy'') = (''xg'')''y'' = (''gx'')''y'' = ''g''(''xy'')}} for each {{math|''g'' ∈ ''G''}}; i.e., {{math|Z(''G'')}} is closed;
# If {{math|''x''}} is in {{math|Z(''G'')}}, then so is {{math|''x''{{sup|−1}}}} as, for all {{math|''g''}} in {{math|''G''}}, {{math|''x''{{sup|−1}}}} commutes with {{math|''g''}}: {{math|1=(''gx'' = ''xg'') ⇒ (''x''{{sup|−1}}''gxx''{{sup|−1}} = ''x''{{sup|−1}}''xgx''{{sup|−1}}) ⇒ (''x''{{sup|−1}}''g'' = ''gx''{{sup|−1}})}}.

Furthermore, the center of {{math|''G''}} is always an [[abelian group|abelian]] and [[normal subgroup]] of {{math|''G''}}. Since all elements of {{math|Z(''G'')}} commute, it is closed under [[conjugate closure|conjugation]].

A group homomorphism {{math|''f'' : ''G'' → ''H''}} might not restrict to a homomorphism between their centers. The image elements  {{math|''f'' (''g'')}} commute with the image {{math|''f'' ( ''G'' )}}, but they need not commute with all of {{math|''H''}} unless {{math|''f''}}  is surjective. Thus the center mapping <math>G\to Z(G)</math> is not a functor between categories Grp and Ab, since it does not induce a map of arrows.