Editing Inner automorphism

{{Short description|Automorphism of a group, ring, or algebra given by the conjugation action of one of its elements}}
In [[abstract algebra]], an '''inner automorphism''' is an [[automorphism]] of a [[Group (mathematics)|group]], [[Ring (mathematics)|ring]], or [[Algebra over a field|algebra]] given by the [[Conjugacy class#Conjugacy as group action|conjugation action]] of a fixed element, called the ''conjugating element''. They can be realized via operations from within the group itself, hence the adjective "inner". These inner automorphisms form a [[subgroup]] of the automorphism group, and the [[Quotient_group|quotient]] of the automorphism group by this subgroup is defined as the [[outer automorphism group]].

==Definition==

If {{mvar|G}} is a group and {{mvar|g}} is an element of {{mvar|G}} (alternatively, if {{mvar|G}} is a ring, and {{mvar|g}} is a [[Unit (ring theory)|unit]]), then the function

:<math>\begin{align}
   \varphi_g\colon G&\to G \\
   \varphi_g(x)&:= g^{-1}xg
 \end{align}</math>

is called '''(right) conjugation by {{mvar|g}}''' (see also [[conjugacy class]]). This function is an [[endomorphism]] of {{mvar|G}}: for all <math>x_1,x_2\in G,</math>

:<math>\varphi_g(x_1 x_2) = g^{-1} x_1 x_2g = g^{-1} x_1 \left(g g^{-1}\right) x_2 g = \left(g^{-1} x_1 g\right)\left(g^{-1} x_2 g\right) = \varphi_g(x_1)\varphi_g(x_2),</math>

where the second equality is given by the insertion of the identity between <math>x_1</math> and <math>x_2.</math> Furthermore, it has a left and right [[inverse function|inverse]], namely <math>\varphi_{g^{-1}}.</math> Thus, <math>\varphi_g</math> is both an [[monomorphism]] and [[epimorphism]], and so an isomorphism of {{mvar|G}} with itself, i.e. an automorphism. An '''inner automorphism''' is any automorphism that arises from conjugation.<ref>{{Cite book |title=Abstract algebra |first1=David S. |last1=Dummit |first2=Richard M. |last2=Foote |date=2004 |publisher=Wiley |isbn=978-0-4714-5234-8 |edition=3rd |location=Hoboken, NJ |page=45 |oclc=248917264}}</ref> 
[[File:Venn Diagram of Homomorphisms.jpg|thumb|General relationship between various group homomorphisms.]]
When discussing right conjugation, the expression <math>g^{-1}xg</math> is often denoted exponentially by <math>x^g.</math> This notation is used because composition of conjugations satisfies the identity: <math>\left(x^{g_1}\right)^{g_2} = x^{g_1g_2}</math> for all <math>g_1, g_2 \in G.</math> This shows that right conjugation gives a right [[group action (mathematics)|action]] of {{mvar|G}} on itself.

A common example is as follows:<ref>{{Cite book |last=Grillet |first=Pierre |title=Abstract Algebra |publisher=Springer |year=2010 |isbn=978-1-4419-2450-6 |edition=2nd |location=New York |pages=56}}</ref><ref>{{Cite book |last=Lang |first=Serge |title=Algebra |publisher=Springer-Verlag |year=2002 |isbn=978-0-387-95385-4 |edition=3rd |location=New York |pages=26}}</ref> 
[[File:Diagram of Inn(G) Example.jpg|thumb|Relationship of morphisms and elements ]]
Describe a homomorphism <math>\Phi</math> for which the image, <math>\text{Im} (\Phi)</math>, is a normal subgroup of inner automorphisms of a group <math>G</math>; alternatively, describe a [[natural homomorphism]] of which the kernel of <math>\Phi</math> is the center of <math>G</math> (all <math>g \in G</math> for which conjugating by them returns the trivial automorphism), in other words, <math>\text{Ker} (\Phi) = \text{Z}(G)</math>. There is always a natural homomorphism <math>\Phi : G \to \text{Aut}(G) </math>, which associates to every <math>g \in G</math> an (inner) automorphism <math>\varphi_{g}</math> in <math>\text{Aut}(G)</math>. Put identically, <math>\Phi : g \mapsto \varphi_{g}</math>.

Let <math>\varphi_{g}(x) := gxg^{-1}</math> as defined above. This requires demonstrating that (1) <math>\varphi_{g}</math> is a homomorphism, (2) <math>\varphi_{g}</math> is also a [[bijection]], (3) <math>\Phi</math> is a homomorphism. 

# <math>\varphi_{g}(xx')=gxx'g^{-1} =gx(g^{-1}g)x'g^{-1} = (gxg^{-1})(gx'g^{-1}) = \varphi_{g}(x)\varphi_{g}(x')</math>
# The condition for bijectivity may be verified by simply presenting an inverse such that we can return to <math>x</math> from <math>gxg^{-1}</math>. In this case it is conjugation by <math>g^{-1}</math>denoted as <math>\varphi_{g^{-1}}</math>. 
# <math>\Phi(gg')(x)=(gg')x(gg')^{-1}</math> and <math>\Phi(g)\circ \Phi(g')(x)=\Phi(g) \circ (g'xg'^{-1}) = gg'xg'^{-1}g^{-1} = (gg')x(gg')^{-1}</math>

==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]].

==Lie algebra case==
An automorphism of a [[Lie algebra]] {{math|𝔊}} is called an inner automorphism if it is of the form {{math|Ad{{sub|''g''}}}}, where {{math|Ad}} is the [[adjoint representation of a Lie group|adjoint map]] and {{mvar|g}} is an element of a [[Lie group]] whose Lie algebra is {{math|𝔊}}.  The notion of inner automorphism for Lie algebras is compatible with the notion for groups in the sense that an inner automorphism of a Lie group induces a unique inner automorphism of the corresponding Lie algebra.

==Extension==
If {{mvar|G}} is the [[group of units]] of a [[ring theory|ring]], {{mvar|A}}, then an inner automorphism on {{mvar|G}} can be extended to a mapping on the [[projective line over a ring|projective line over {{mvar|A}}]] by the group of units of the [[matrix ring]], {{math|M{{sub|2}}(''A'')}}. In particular, the inner automorphisms of the [[classical group]]s can be extended in that way.

==References==
{{reflist}}

== Further reading ==
* {{citation | mr=2574864  | last1=Abdollahi | first1=A. | title=Powerful ''p''-groups have non-inner automorphisms of order ''p'' and some cohomology | journal=J. Algebra | volume=323  | year=2010 | issue=3 | pages=779&ndash;789 | doi=10.1016/j.jalgebra.2009.10.013| arxiv=0901.3182 }}
* {{citation | mr=2333188  | last1=Abdollahi | first1=A. | title=Finite ''p''-groups of class ''2'' have noninner automorphisms of order ''p'' | journal=J. Algebra | volume=312  | year=2007 | issue=2 | pages=876&ndash;879 | doi=10.1016/j.jalgebra.2006.08.036| arxiv=math/0608581 }}
* {{citation | mr=1898386  | last1=Deaconescu | first1=M. | last2=Silberberg | first2=G. | title=Noninner automorphisms of order ''p'' of finite ''p''-groups | journal=J. Algebra | volume=250  | year=2002 | pages=283&ndash;287 | doi=10.1006/jabr.2001.9093| doi-access=free }}
* {{citation | mr=0193144  | last1=Gaschütz | first1=W. | title=Nichtabelsche ''p''-Gruppen besitzen äussere ''p''-Automorphismen| journal=J. Algebra | volume=4  | year=1966 | pages=1&ndash;2 | doi=10.1016/0021-8693(66)90045-7| doi-access=free }}
* {{citation | mr=0173708  | last1=Liebeck | first1=H. | title=Outer automorphisms in nilpotent ''p''-groups of class ''2'' | journal=J. London Math. Soc. | volume=40  | year=1965 | pages=268&ndash;275 | doi=10.1112/jlms/s1-40.1.268}}
*{{springer|title=Inner automorphism|id=I/i051230|last=Remeslennikov|first=V.N.}}
*{{MathWorld|title=Inner Automorphism|urlname=InnerAutomorphism}}

{{DEFAULTSORT:Inner Automorphism}}
[[Category:Group theory]]
[[Category:Group automorphisms]]

[[de:Automorphismus#Innere Automorphismen]]