Editing Inner automorphism (section)

===Types of groups===
The inner automorphism group of a group {{mvar|G}}, {{math|Inn(''G'')}}, is trivial (i.e., consists only of the [[identity element]]) [[if and only if]] {{mvar|G}} is [[abelian group|abelian]].

The group {{math|Inn(''G'')}} is [[cyclic group|cyclic]] only when it is trivial.

At the opposite end of the spectrum, the inner automorphisms may exhaust the entire automorphism group; a group whose automorphisms are all inner and whose center is trivial is called [[complete group|complete]]. This is the case for all of the symmetric groups on {{mvar|n}} elements when {{mvar|n}} is not 2 or 6.  When {{math|''n'' {{=}} 6}}, the [[symmetric group]] has a unique non-trivial class of non-inner automorphisms, and when {{math|''n'' {{=}} 2}}, the symmetric group, despite having no non-inner automorphisms, is abelian, giving a non-trivial center, disqualifying it from being complete.

If the inner automorphism group of a [[perfect group]] {{mvar|G}} is simple, then {{mvar|G}} is called [[quasisimple group|quasisimple]].