Editing Inner automorphism (section)

==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>