inner automorphism
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In
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
, an inner automorphism is an
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphism ...
of a group, ring, or
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
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 In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
of the automorphism group, and the
quotient In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...
of the automorphism group by this subgroup is defined as the
outer automorphism group In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has ...
.


Definition

If is a group and is an element of (alternatively, if is a ring, and is a unit), then the function :\begin \varphi_g\colon G&\to G \\ \varphi_g(x)&:= g^xg \end is called (right) conjugation by (see also
conjugacy class In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other ...
). This function is an
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
of : for all x_1,x_2\in G, :\varphi_g(x_1 x_2) = g^ x_1 x_2g = g^ x_1 \left(g g^\right) x_2 g = \left(g^ x_1 g\right)\left(g^ 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 x_1 and x_2. Furthermore, it has a left and right inverse, namely \varphi_. Thus, \varphi_g is both an
monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphis ...
and
epimorphism In category theory, an epimorphism is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f \implies g_1 = g_2. Epimorphisms are categorical analo ...
, and so an isomorphism of with itself, i.e. an automorphism. An inner automorphism is any automorphism that arises from conjugation. When discussing right conjugation, the expression g^xg is often denoted exponentially by x^g. This notation is used because composition of conjugations satisfies the identity: \left(x^\right)^ = x^ for all g_1, g_2 \in G. This shows that right conjugation gives a right action of on itself. A common example is as follows: Describe a homomorphism \Phi for which the image, \text (\Phi), is a normal subgroup of inner automorphisms of a group G; alternatively, describe a natural homomorphism of which the kernel of \Phi is the center of G (all g \in G for which conjugating by them returns the trivial automorphism), in other words, \text (\Phi) = \text(G). There is always a natural homomorphism \Phi : G \to \text(G) , which associates to every g \in G an (inner) automorphism \varphi_ in \text(G). Put identically, \Phi : g \mapsto \varphi_. Let \varphi_(x) := gxg^ as defined above. This requires demonstrating that (1) \varphi_ is a homomorphism, (2) \varphi_ is also a
bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
, (3) \Phi is a homomorphism. # \varphi_(xx')=gxx'g^ =gx(g^g)x'g^ = (gxg^)(gx'g^) = \varphi_(x)\varphi_(x') # The condition for bijectivity may be verified by simply presenting an inverse such that we can return to x from gxg^. In this case it is conjugation by g^denoted as \varphi_. # \Phi(gg')(x)=(gg')x(gg')^ and \Phi(g)\circ \Phi(g')(x)=\Phi(g) \circ (g'xg'^) = gg'xg'^g^ = (gg')x(gg')^


Inner and outer automorphism groups

The
composition Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography * Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include ...
of two inner automorphisms is again an inner automorphism, and with this operation, the collection of all inner automorphisms of is a group, the inner automorphism group of denoted . is a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...
of the full
automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
of . The
outer automorphism group In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has ...
, is the
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored out"). For ex ...
:\operatorname(G) = \operatorname(G) / \operatorname(G). The outer automorphism group measures, in a sense, how many automorphisms of are not inner. Every non-inner automorphism yields a non-trivial element of , but different non-inner automorphisms may yield the same element of . Saying that conjugation of by leaves unchanged is equivalent to saying that and commute: :a^xa = x \iff xa = ax. Therefore the existence and number of inner automorphisms that are not the
identity mapping Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
is a kind of measure of the failure of the
commutative law In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a p ...
in the group (or ring). An automorphism of a group is inner if and only if it extends to every group containing . By associating the element with the inner automorphism in as above, one obtains an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
between the
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored out"). For ex ...
(where is the center of ) and the inner automorphism group: :G\,/\,\mathrm(G) \cong \operatorname(G). This is a consequence of the
first isomorphism theorem In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship among quotients, homomorphisms, and subobjects. Versions of the theorems exist for ...
, because is precisely the set of those elements of that give the identity mapping as corresponding inner automorphism (conjugation changes nothing).


Non-inner automorphisms of finite -groups

A result of Wolfgang Gaschütz says that if is a finite non-abelian -group, then has an automorphism of -power order which is not inner. It is an
open problem In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (i.e., no solution for it is kno ...
whether every non-abelian -group has an automorphism of order . The latter question has positive answer whenever has one of the following conditions: # is nilpotent of class 2 # is a regular -group # is a powerful -group # The
centralizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set \operatorname_G(S) of elements of ''G'' that commute with every element of ''S'', or equivalently, the set of ele ...
in , , of the center, , of the
Frattini subgroup In mathematics, particularly in group theory, the Frattini subgroup \Phi(G) of a group is the intersection of all maximal subgroups of . For the case that has no maximal subgroups, for example the trivial group or a Prüfer group, it is def ...
, , of , , is not equal to


Types of groups

The inner automorphism group of a group , , is trivial (i.e., consists only of the
identity element In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
)
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
is abelian. The group 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 elements when is not 2 or 6. When , the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
has a unique non-trivial class of non-inner automorphisms, and when , 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 In mathematics, more specifically in group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients. Examples The smallest (non-trivial) perfect group ...
is simple, then is called quasisimple.


Lie algebra case

An automorphism of a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
is called an inner automorphism if it is of the form , where is the adjoint map and is an element of a
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
whose Lie algebra is . 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 is the
group of units In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that vu = uv = 1, where is the multiplicative identity; the ele ...
of a
ring (The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...
, , then an inner automorphism on can be extended to a mapping on the projective line over by the group of units of the matrix ring, . In particular, the inner automorphisms of the
classical group In mathematics, the classical groups are defined as the special linear groups over the reals \mathbb, the complex numbers \mathbb and the quaternions \mathbb together with special automorphism groups of Bilinear form#Symmetric, skew-symmetric an ...
s can be extended in that way.


References


Further reading

* * * * * * * {{DEFAULTSORT:Inner Automorphism Group theory Group automorphisms de:Automorphismus#Innere Automorphismen