mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 iden ...

is said to be complete if every

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 is inner, and it is centerless; that is, it has a trivial

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 ...

and trivial center. Equivalently, a group is complete if the

conjugation Conjugation or conjugate may refer to: Linguistics *Grammatical conjugation, the modification of a verb from its basic form *Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics *Complex conjugation, the change o ...

map, (sending an element to conjugation by ), is 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 ...

injectivity In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...

implies that only conjugation by 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 ...

is the identity automorphism, meaning the group is centerless, while surjectivity implies it has no outer automorphisms.

Examples

As an example, all 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 ...

s, , are complete except when . For the case , the group has a non-trivial center, while for the case , there is an

outer automorphism 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 a ...

. The automorphism group of a

simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...

is an

almost simple group In mathematics, a group (mathematics), group is said to be almost simple if it contains a non-abelian group, abelian simple group and is contained within the automorphism group of that simple group – that is, if it fits between a (non-abelian) sim ...

; for a non- abelian simple group , the automorphism group of is complete.

Properties

A complete group is always

isomorphic 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 ...

to its

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 ...

(via sending an element to conjugation by that element), although the converse need not hold: for example, the

dihedral group In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest example ...

of 8 elements is isomorphic to its automorphism group, but it is not complete. For a discussion, see .

Extensions of complete groups

Assume that a group is a group extension given as a

short exact sequence In mathematics, an exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Im ...

of groups : with kernel, , and quotient, . If the kernel, , is a complete group then the extension splits: is isomorphic to the

direct product In mathematics, a direct product of objects already known can often be defined by giving a new one. That induces a structure on the Cartesian product of the underlying sets from that of the contributing objects. The categorical product is an abs ...

, . A proof using

homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...

s and exact sequences can be given in a natural way: The action of (by

) on the

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 ...

, , gives rise to a group homomorphism, . Since and has trivial center the homomorphism is surjective and has an obvious section given by the inclusion of in . The kernel of is 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 ...

of in , and so is at least a

semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. It is usually denoted with the symbol . There are two closely related concepts of semidirect product: * an ''inner'' sem ...

, , but the action of on is trivial, and so the product is direct. This can be restated in terms of elements and internal conditions: If is a normal, complete

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 a group , then is a direct product. The proof follows directly from the definition: is centerless giving is trivial. If is an element of then it induces an automorphism of by conjugation, but and this conjugation must be equal to conjugation by some element of . Then conjugation by is the identity on and so is in and every element, , of is a product in .

References

* * {{Citation , last1=Rotman , first1=Joseph J. , title=An introduction to the theory of groups , publisher=

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...

, location=Berlin, New York , isbn=978-0-387-94285-8 , year=1994 (chapter 7, in particular theorems 7.15 and 7.17).

External links

Joel David Hamkins Joel David Hamkins is an American mathematician and philosopher who is the John Cardinal O'Hara Professor of Logic at the University of Notre Dame. He has made contributions in mathematical logic, mathematical and philosophical logic, set theor ...

How tall is the automorphism tower of a group?
Properties of groups