Editing Inner automorphism (section)

==Inner and outer automorphism groups==
The [[functional composition|composition]] of two inner automorphisms is again an inner automorphism, and with this operation, the collection of all inner automorphisms of {{mvar|G}} is a group, the inner automorphism group of {{mvar|G}} denoted {{math|Inn(''G'')}}.

{{math|Inn(''G'')}} is a [[normal subgroup]] of the full [[automorphism group]] {{math|Aut(''G'')}} of {{mvar|G}}. The [[outer automorphism group]], {{math|Out(''G'')}} is the [[quotient group]]
:<math>\operatorname{Out}(G) = \operatorname{Aut}(G) / \operatorname{Inn}(G).</math>

The outer automorphism group measures, in a sense, how many automorphisms of {{mvar|G}} are not inner.  Every non-inner automorphism yields a non-trivial element of {{math|Out(''G'')}}, but different non-inner automorphisms may yield the same element of {{math|Out(''G'')}}.

Saying that conjugation of {{mvar|x}} by {{mvar|a}} leaves {{mvar|x}} unchanged is equivalent to saying that {{mvar|a}} and {{mvar|x}} commute:
:<math>a^{-1}xa = x \iff xa = ax.</math>

Therefore the existence and number of inner automorphisms that are not the [[identity mapping]] is a kind of measure of the failure of the [[commutative law]] in the group (or ring).

An automorphism of a group {{mvar|G}} is inner if and only if it extends to every group containing {{mvar|G}}.<ref>{{Citation|title=A characterization of inner automorphisms|year=1987|last1=Schupp|first1=Paul E.|author-link1=Paul Schupp|journal=Proceedings of the American Mathematical Society|volume=101|issue=2|pages=226–228|publisher=American Mathematical Society|doi=10.2307/2045986|doi-access=free|jstor=2045986|mr=902532|url=https://www.ams.org/journals/proc/1987-101-02/S0002-9939-1987-0902532-X/S0002-9939-1987-0902532-X.pdf}}</ref>

By associating the element {{math|''a'' ∈ ''G''}} with the inner automorphism {{math|''f''(''x'') {{=}} ''x''{{sup|''a''}}}} in {{math|Inn(''G'')}} as above, one obtains an [[group isomorphism|isomorphism]] between the [[quotient group]] {{math|''G'' / Z(''G'')}} (where {{math|Z(''G'')}} is the [[center of a group|center]] of {{mvar|G}}) and the inner automorphism group:
:<math>G\,/\,\mathrm{Z}(G) \cong \operatorname{Inn}(G).</math>

This is a consequence of the [[isomorphism theorem|first isomorphism theorem]], because {{math|Z(''G'')}} is precisely the set of those elements of {{mvar|G}} that give the identity mapping as corresponding inner automorphism (conjugation changes nothing).

===Non-inner automorphisms of finite {{mvar|p}}-groups===
A result of Wolfgang Gaschütz says that if {{mvar|G}} is a finite non-abelian [[p-group|{{mvar|p}}-group]], then {{mvar|G}} has an automorphism of {{mvar|p}}-power order which is not inner.

It is an [[open problem]] whether every non-abelian {{mvar|p}}-group {{mvar|G}} has an automorphism of order {{mvar|p}}. The latter question has positive answer whenever {{mvar|G}} has one of the following conditions:
# {{mvar|G}} is nilpotent of class 2
# {{mvar|G}} is a [[regular p-group|regular {{mvar|p}}-group]]
# {{math|''G'' / Z(''G'')}} is a [[powerful p-group|powerful {{mvar|p}}-group]]
# The [[centralizer and normalizer|centralizer]] in {{mvar|G}}, {{math|''C''{{sub|''G''}}}}, of the center, {{mvar|Z}}, of the [[Frattini subgroup]], {{math|Φ}}, of {{mvar|G}}, {{math|''C''{{sub|''G''}} ∘ ''Z'' ∘ Φ(''G'')}}, is not equal to {{math|Φ(''G'')}}

===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]].