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In
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
an inner automorphism is an
automorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ... of a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
, ring, or
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...
given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple 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 the definition of the
outer automorphism groupIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
.

# Definition

If is a group and is an element of (alternatively, if is a ring, and is a
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * Unit (album), ...
), then the function :$\begin \varphi_g\colon G&\to G \\ \varphi_g\left(x\right)&:= g^xg \end$ is called (right) conjugation by (see also
conjugacy class In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
). 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 group ... of : for all $x_1,x_2\in G,$ :$\varphi_g\left(x_1 x_2\right) = g^ x_1 x_2g = \left\left(g^ x_1 g\right\right)\left\left(g^ x_2 g\right\right) = \varphi_g\left(x_1\right)\varphi_g\left(x_2\right),$ 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 Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when add ...
, namely $\varphi_.$ Thus, $\varphi_g$ is
bijective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... , 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: $\left\left(x^\right\right)^ = x^$ for all $g_1, g_2 \in G.$ This shows that conjugation gives a right
action ACTION is a bus operator in , Australia owned by the . History On 19 July 1926, the commenced operating public bus services between Eastlake (now ) in the south and in the north. The service was first known as Canberra City Omnibus Se ...
of on itself.

# Inner and outer automorphism groups

The
composition Composition or Compositions may refer to: Arts * Composition (dance), practice and teaching of choreography * Composition (music), an original piece of music and its creation *Composition (visual arts) The term composition means "putting togethe ...
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 In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
of the full
automorphism group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
of . The
outer automorphism groupIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, is the
quotient group A quotient group or factor group is a mathematical group (mathematics), 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 "factore ...
:$\operatorname\left(G\right) = \operatorname\left(G\right) / \operatorname\left(G\right).$ 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: :$a^xa = x \iff ax = xa.$ 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 (mathematics), function that always returns the same value that was ...
is a kind of measure of the failure of the
commutative law In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
between the
quotient group A quotient group or factor group is a mathematical group (mathematics), 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 "factore ...
(where is the
center Center or centre may refer to: Mathematics *Center (geometry) In geometry, a centre (or center) (from Ancient Greek language, Greek ''κέντρον'') of an object is a point in some sense in the middle of the object. According to the speci ...
of ) and the inner automorphism group: :$G/Z\left(G\right) \cong \operatorname\left(G\right).$ 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 between 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 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
in , , of the center, , of the Frattini subgroup, , 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
)
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
is abelian. The group is
cyclic Cycle or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in social scienc ... 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 has a unique non-trivial class of outer automorphisms, and when , the symmetric group, despite having no outer automorphisms, is abelian, giving a non-trivial center, disqualifying it from being complete. If the inner automorphism group of a
perfect group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
is simple, then is called quasisimple.

# Lie algebra case

An automorphism of a
Lie algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
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 the branch of abstract algebra known as ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations def ...
of a ring, , 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 abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
, . In particular, the inner automorphisms of the
classical group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
s can be extended in that way.