Inner automorphism

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In abstract algebra, an inner automorphism is an automorphism of a group, ring, or algebra given by the 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 of the automorphism group by this subgroup is defined as the outer automorphism group.

Definition

If $G$ is a group and $g$ is an element of $G$ (alternatively, if $G$ is a ring, and $g$ is a unit), then the function

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \varphi_g\colon G&\to G \\ \varphi_g(x)&:= g^{-1}xg \end{align}}

is called (right) conjugation by $g$ (see also conjugacy class). This function is an endomorphism of $G$ : for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1,x_2\in G,}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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),}

where the second equality is given by the insertion of the identity between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_2.} Furthermore, it has a left and right inverse, namely Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi_{g^{-1}}.} Thus, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi_g} is both an monomorphism and epimorphism, and so an isomorphism of $G$ with itself, i.e. an automorphism. An inner automorphism is any automorphism that arises from conjugation.^[1]

File:Venn Diagram of Homomorphisms.jpg

General relationship between various group homomorphisms.

When discussing right conjugation, the expression Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g^{-1}xg} is often denoted exponentially by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^g.} This notation is used because composition of conjugations satisfies the identity: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(x^{g_1}\right)^{g_2} = x^{g_1g_2}} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_1, g_2 \in G.} This shows that right conjugation gives a right action of $G$ on itself.

A common example is as follows:^[2]^[3]

File:Diagram of Inn(G) Example.jpg

Relationship of morphisms and elements

Describe a homomorphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi} for which the image, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Im} (\Phi)} , is a normal subgroup of inner automorphisms of a group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} ; alternatively, describe a natural homomorphism of which the kernel of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi} is the center of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} (all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \in G} for which conjugating by them returns the trivial automorphism), in other words, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Ker} (\Phi) = \text{Z}(G)} . There is always a natural homomorphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi : G \to \text{Aut}(G) } , which associates to every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \in G} an (inner) automorphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi_{g}} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Aut}(G)} . Put identically, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi : g \mapsto \varphi_{g}} .

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi_{g}(x) := gxg^{-1}} as defined above. This requires demonstrating that (1) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi_{g}} is a homomorphism, (2) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi_{g}} is also a bijection, (3) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi} is a homomorphism.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi_{g}(xx')=gxx'g^{-1} =gx(g^{-1}g)x'g^{-1} = (gxg^{-1})(gx'g^{-1}) = \varphi_{g}(x)\varphi_{g}(x')}
The condition for bijectivity may be verified by simply presenting an inverse such that we can return to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle gxg^{-1}} . In this case it is conjugation by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g^{-1}} denoted as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi_{g^{-1}}} .
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi(gg')(x)=(gg')x(gg')^{-1}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi(g)\circ \Phi(g')(x)=\Phi(g) \circ (g'xg'^{-1}) = gg'xg'^{-1}g^{-1} = (gg')x(gg')^{-1}}

Inner and outer automorphism groups

The composition of two inner automorphisms is again an inner automorphism, and with this operation, the collection of all inner automorphisms of $G$ is a group, the inner automorphism group of $G$ denoted $Inn(G)$ .

$Inn(G)$ is a normal subgroup of the full automorphism group $Aut(G)$ of $G$ . The outer automorphism group, $Out(G)$ is the quotient group

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Out}(G) = \operatorname{Aut}(G) / \operatorname{Inn}(G).}

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

Saying that conjugation of $x$ by $a$ leaves $x$ unchanged is equivalent to saying that $a$ and $x$ commute:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-1}xa = x \iff xa = ax.}

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 $G$ is inner if and only if it extends to every group containing $G$ .^[4]

By associating the element $a \in G$ with the inner automorphism $f (x) = x a$ in $Inn(G)$ as above, one obtains an isomorphism between the quotient group $G / Z(G)$ (where $Z(G)$ is the center of $G$ ) and the inner automorphism group:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G\,/\,\mathrm{Z}(G) \cong \operatorname{Inn}(G).}

This is a consequence of the first isomorphism theorem, because $Z(G)$ is precisely the set of those elements of $G$ that give the identity mapping as corresponding inner automorphism (conjugation changes nothing).

Non-inner automorphisms of finite $p$ -groups

A result of Wolfgang Gaschütz says that if $G$ is a finite non-abelian $p$ -group, then $G$ has an automorphism of $p$ -power order which is not inner.

It is an open problem whether every non-abelian $p$ -group $G$ has an automorphism of order $p$ . The latter question has positive answer whenever $G$ has one of the following conditions:

$G$ is nilpotent of class 2
$G$ is a regular $p$ -group
$G / Z(G)$ is a powerful $p$ -group
The centralizer in $G$ , $C G$ , of the center, $Z$ , of the Frattini subgroup, $Φ$ , of $G$ , $C G \circ Z \circ Φ(G)$ , is not equal to $Φ(G)$

Types of groups

The inner automorphism group of a group $G$ , $Inn(G)$ , is trivial (i.e., consists only of the identity element) if and only if $G$ is abelian.

The group $Inn(G)$ is 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. This is the case for all of the symmetric groups on $n$ elements when $n$ is not 2 or 6. When $n = 6$ , the symmetric group has a unique non-trivial class of non-inner automorphisms, and when $n = 2$ , 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 $G$ is simple, then $G$ is called quasisimple.

Lie algebra case

An automorphism of a Lie algebra $𝔊$ is called an inner automorphism if it is of the form $Ad g$ , where $Ad$ is the adjoint map and $g$ is an element of a Lie group whose Lie algebra is $𝔊$ . 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 $G$ is the group of units of a ring, $A$ , then an inner automorphism on $G$ can be extended to a mapping on the projective line over $A$ by the group of units of the matrix ring, $M 2 (A)$ . In particular, the inner automorphisms of the classical groups can be extended in that way.

References

↑ Dummit, David S.; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. p. 45. ISBN 978-0-4714-5234-8. OCLC 248917264.
↑ Grillet, Pierre (2010). Abstract Algebra (2nd ed.). New York: Springer. p. 56. ISBN 978-1-4419-2450-6.
↑ Lang, Serge (2002). Algebra (3rd ed.). New York: Springer-Verlag. p. 26. ISBN 978-0-387-95385-4.
↑ Schupp, Paul E. (1987), "A characterization of inner automorphisms" (PDF), Proceedings of the American Mathematical Society, American Mathematical Society, 101 (2): 226–228, doi:10.2307/2045986, JSTOR 2045986, MR 0902532

Inner automorphism

Contents

Definition

Inner and outer automorphism groups

Non-inner automorphisms of finite $p$ -groups

Types of groups

Lie algebra case

Extension

References

Further reading

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Inner automorphism

Definition

Inner and outer automorphism groups

Non-inner automorphisms of finite p-groups

Types of groups

Lie algebra case

Extension

References

Further reading

Navigation menu

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Non-inner automorphisms of finite $p$ -groups