In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ' ...
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 automorphis ...
of a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
,
ring
Ring 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
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
, or
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
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 defined as the
outer automorphism group.
Definition
If is a group and is an element of (alternatively, if is a ring, and is a
unit), then the function
:
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 wo ...
). This function is an
endomorphism of : for all
:
where the second equality is given by the insertion of the identity between
and
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 a ...
, namely
Thus,
is
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
, 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
is often denoted exponentially by
This notation is used because composition of conjugations satisfies the identity:
for all
This shows that right conjugation gives a right
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
of on itself.
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 v ...
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 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, is the
quotient group
:
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:
:
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 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. Most familiar as the name ...
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 between the
quotient group (where is the
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentricity ...
of ) and the inner automorphism group:
:
This is a consequence of the
first isomorphism theorem, 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, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', ...
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, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...
)
if and only if is
abelian
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
.
The group is
cyclic
Cycle, cycles, 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 so ...
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
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
. 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 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 is simple, then is called
quasisimple
In mathematics, a quasisimple group (also known as a covering group) is a group that is a perfect central extension ''E'' of a simple group ''S''. In other words, there is a short exact sequence
:1 \to Z(E) \to E \to S \to 1
such that E = , ...
.
Lie algebra case
An automorphism of a
Lie algebra 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 that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the ad ...
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 of a
ring
Ring 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
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
, , 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 groups can be extended in that way.
References
Further reading
*
*
*
*
*
*
*
{{DEFAULTSORT:Inner Automorphism
Group theory
Group automorphisms
de:Automorphismus#Innere Automorphismen