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''of a [[Group|group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051230/i0512301.png" />''
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''of a [[group]] $G$''
  
An [[Automorphism|automorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051230/i0512302.png" /> such that
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An [[automorphism]] $\phi$ such that
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$$
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\phi(x) = g^{-1} x g
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051230/i0512303.png" /></td> </tr></table>
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for a certain fixed element $g \in G$: that is, $\phi$ is [[Conjugate elements|conjugation]] by $g$. The set of all inner automorphisms of $G$ forms a normal subgroup $\mathrm{Inn}(G)$ in the group $\mathrm{Aut}(G)$ of all automorphisms of $G$; this subgroup is isomorphic to $G / Z(G)$, where $Z(G)$ is the centre of $G$ (cf. [[Centre of a group|Centre of a group]]). Automorphisms that are not inner are called ''outer automorphisms''.  The ''outer automorphism group'' is the quotient $\mathrm{Out}(G) = \mathrm{Aut}(G)/\mathrm{Inn}(G)$.
 
 
for a certain fixed element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051230/i0512304.png" />. The set of all inner automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051230/i0512305.png" /> forms a normal subgroup in the group of all automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051230/i0512306.png" />; this subgroup is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051230/i0512307.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051230/i0512308.png" /> is the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051230/i0512309.png" /> (cf. [[Centre of a group|Centre of a group]]). Automorphisms that are not inner are called outer automorphisms.
 
  
 
Other relevant concepts include those of an inner automorphism of a monoid (a semi-group with a unit element) and an inner automorphism of a ring (associative with a unit element), which are introduced in a similar way using invertible elements.
 
Other relevant concepts include those of an inner automorphism of a monoid (a semi-group with a unit element) and an inner automorphism of a ring (associative with a unit element), which are introduced in a similar way using invertible elements.
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====Comments====
 
====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051230/i05123010.png" /> be a [[Lie algebra|Lie algebra]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051230/i05123011.png" /> an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051230/i05123012.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051230/i05123013.png" /> is nilpotent. Then
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Let $\mathfrak{g}$ be a [[Lie algebra]] and $x \in \mathfrak{g}$ an element of $\mathfrak{g}$ for which $\mathrm{ad}(x) : y \mapsto [x,y]$ is nilpotent. Then
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051230/i05123014.png" /></td> </tr></table>
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\exp(\mathrm{ad}(x)) = \mathrm{id} + \mathrm{ad}(x) + \frac{1}{2!}\mathrm{ad}(x)^2 + \cdots
 
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$$
defines an automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051230/i05123015.png" />. Such an automorphism is called an inner automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051230/i05123016.png" />. More generally, the elements in the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051230/i05123017.png" /> generated by them are called inner automorphisms. It is a normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051230/i05123018.png" />.
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defines an automorphism of $\mathfrak{g}$. Such an automorphism is called an inner automorphism of $\mathfrak{g}$. More generally, the elements in the group $\mathrm{Int}(\mathfrak{g})$ generated by them are called inner automorphisms. It is a normal subgroup of $\mathrm{Aut}(\mathfrak{g})$.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051230/i05123019.png" /> is a real or complex [[Lie group|Lie group]] with semi-simple Lie algebra, then the inner automorphisms constitute precisely the identity component of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051230/i05123020.png" /> of automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051230/i05123021.png" />.
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If $G$ is a real or complex [[Lie group]] with semi-simple Lie algebra, then the inner automorphisms constitute precisely the identity component of the group $\mathrm{Aut}(\mathfrak{g})$ of automorphisms of $\mathfrak{g}$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Hall jr.,  "The theory of groups" , Macmillan  (1959)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.E. Humphreys,  "Introduction to Lie algebras and representation theory" , Springer  (1972)  pp. §5.4</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.-P. Serre,  "Algèbres de Lie semi-simples complexes" , Benjamin  (1966)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Hall jr.,  "The theory of groups" , Macmillan  (1959)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  J.E. Humphreys,  "Introduction to Lie algebras and representation theory" , Springer  (1972)  pp. §5.4</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  J.-P. Serre,  "Algèbres de Lie semi-simples complexes" , Benjamin  (1966)</TD></TR>
 +
</table>

Latest revision as of 21:00, 29 November 2014

of a group $G$

An automorphism $\phi$ such that $$ \phi(x) = g^{-1} x g $$

for a certain fixed element $g \in G$: that is, $\phi$ is conjugation by $g$. The set of all inner automorphisms of $G$ forms a normal subgroup $\mathrm{Inn}(G)$ in the group $\mathrm{Aut}(G)$ of all automorphisms of $G$; this subgroup is isomorphic to $G / Z(G)$, where $Z(G)$ is the centre of $G$ (cf. Centre of a group). Automorphisms that are not inner are called outer automorphisms. The outer automorphism group is the quotient $\mathrm{Out}(G) = \mathrm{Aut}(G)/\mathrm{Inn}(G)$.

Other relevant concepts include those of an inner automorphism of a monoid (a semi-group with a unit element) and an inner automorphism of a ring (associative with a unit element), which are introduced in a similar way using invertible elements.


Comments

Let $\mathfrak{g}$ be a Lie algebra and $x \in \mathfrak{g}$ an element of $\mathfrak{g}$ for which $\mathrm{ad}(x) : y \mapsto [x,y]$ is nilpotent. Then $$ \exp(\mathrm{ad}(x)) = \mathrm{id} + \mathrm{ad}(x) + \frac{1}{2!}\mathrm{ad}(x)^2 + \cdots $$ defines an automorphism of $\mathfrak{g}$. Such an automorphism is called an inner automorphism of $\mathfrak{g}$. More generally, the elements in the group $\mathrm{Int}(\mathfrak{g})$ generated by them are called inner automorphisms. It is a normal subgroup of $\mathrm{Aut}(\mathfrak{g})$.

If $G$ is a real or complex Lie group with semi-simple Lie algebra, then the inner automorphisms constitute precisely the identity component of the group $\mathrm{Aut}(\mathfrak{g})$ of automorphisms of $\mathfrak{g}$.

References

[a1] M. Hall jr., "The theory of groups" , Macmillan (1959)
[a2] J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972) pp. §5.4
[a3] J.-P. Serre, "Algèbres de Lie semi-simples complexes" , Benjamin (1966)
How to Cite This Entry:
Inner automorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inner_automorphism&oldid=15849
This article was adapted from an original article by V.N. Remeslennikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article