Inner automorphism

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

\begin{aligned} φ_{g} : G & \to G \\ φ_{g} (x) & : = g^{- 1} x g \end{aligned}

is called (right) conjugation by $g$ (see also conjugacy class). This function is an endomorphism of $G$ : for all $x_{1}, x_{2} \in G,$

φ_{g} (x_{1} x_{2}) = g^{- 1} x_{1} x_{2} g = g^{- 1} x_{1} (g g^{- 1}) x_{2} g = (g^{- 1} x_{1} g) (g^{- 1} x_{2} g) = φ_{g} (x_{1}) φ_{g} (x_{2}),

where the second equality is given by the insertion of the identity between $x_{1}$ and $x_{2} .$ Furthermore, it has a left and right inverse, namely $φ_{g^{- 1}} .$ Thus, $φ_{g}$ 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 homomorphisms.

When discussing right conjugation, the expression $g^{- 1} x g$ is often denoted exponentially by $x^{g} .$ This notation is used because composition of conjugations satisfies the identity: ${(x^{g_{1}})}^{g_{2}} = x^{g_{1} g_{2}}$ for all $g_{1}, g_{2} \in G .$ 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 $Φ$ for which the image, $Im (Φ)$ , is a normal subgroup of inner automorphisms of a group $G$ ; alternatively, describe a natural homomorphism of which the kernel of $Φ$ is the center of $G$ (all $g \in G$ for which conjugating by them returns the trivial automorphism), in other words, $Ker (Φ) = Z (G)$ . There is always a natural homomorphism $Φ : G \to Aut (G)$ , which associates to every $g \in G$ an (inner) automorphism $φ_{g}$ in $Aut (G)$ . Put identically, $Φ : g \mapsto φ_{g}$ . Let $φ_{g} (x) : = g x g^{- 1}$ as defined above. This requires demonstrating that (1) $φ_{g}$ is a homomorphism, (2) $φ_{g}$ is also a bijection, (3) $Φ$ is a homomorphism.

$φ_{g} (x x^{'}) = g x x^{'} g^{- 1} = g x (g^{- 1} g) x^{'} g^{- 1} = (g x g^{- 1}) (g x^{'} g^{- 1}) = φ_{g} (x) φ_{g} (x^{'})$
The condition for bijectivity may be verified by simply presenting an inverse such that we can return to $x$ from $g x g^{- 1}$ . In this case it is conjugation by $g^{- 1}$ denoted as $φ_{g^{- 1}}$ .
$Φ (g g^{'}) (x) = (g g^{'}) x (g g^{'})^{- 1}$ and $Φ (g) \circ Φ (g^{'}) (x) = Φ (g) \circ (g^{'} h g'^{- 1}) = g g^{'} h g'^{- 1} g^{- 1} = (g g^{'}) h (g g^{'})^{- 1}$

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

Out (G) = Aut (G) / 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:

a^{- 1} x a = x ⟺ x a = a x .

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:

G / Z (G) ≅ 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, 101 (2), American Mathematical Society: 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

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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

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Non-inner automorphisms of finite $p$ -groups