In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, 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 ...
is said to be almost simple if it contains a non-
abelian 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 ...
and is contained within the
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 that simple group – that is, if it fits between a (non-abelian) simple group and its automorphism group. In symbols, a group ''A'' is almost simple if there is a (non-abelian) simple group ''S'' such that
Examples
* Trivially, non-abelian simple groups and the full group of automorphisms are almost simple, but proper examples exist, meaning almost simple groups that are neither simple nor the full automorphism group.
* For
or
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 group \m ...
is the automorphism group of the simple
alternating group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or
Basic prop ...
so
is almost simple in this trivial sense.
* For
there is a proper example, as
sits properly between the simple
and
due to the
exceptional outer automorphism of
Two other groups, the
Mathieu group
In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 obje ...
and the
projective general linear group
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associat ...
also sit properly between
and
Properties
The full automorphism group of a non-abelian simple group is a
complete group In mathematics, a group is said to be complete if every automorphism of is inner, and it is centerless; that is, it has a trivial outer automorphism group and trivial center.
Equivalently, a group is complete if the conjugation map, (sending ...
(the conjugation map is an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping 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 them. The word is ...
to the automorphism group), but proper
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
s of the full automorphism group need not be complete.
Structure
By the
Schreier conjecture In finite group theory, the Schreier conjecture asserts that the outer automorphism group of every finite simple group is solvable. It was proposed by Otto Schreier in 1926, and is now known to be true as a result of the classification of finite si ...
, now generally accepted as a
corollary of the
classification of finite simple groups
In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else it ...
, the outer automorphism group of a
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
simple group is a
solvable group
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates ...
. Thus a finite almost simple group is an extension of a solvable group by a simple group.
See also
*
Quasisimple group
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 = , E/ ...
*
Semisimple group
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direc ...
Notes
{{reflist, group=note
External links
Almost simple groupat the Group Properties wiki
Properties of groups