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 quasisimple group (also known as a covering group) is 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 ...

that is a perfect central extension ''E'' 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 ...

''S''. In other words, there is 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 ...

1 \to Z(E) \to E \to S \to 1

such that

E =

, E The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...

/math>, where

Z(E)

denotes the center of ''E'' and , denotes the

commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...

. I. Martin Isaacs, ''Finite group theory'' (2008), p. 272. Equivalently, a group is quasisimple if it is equal to its

commutator subgroup In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal ...

and its

inner automorphism group In abstract algebra, an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via operations from within the group itself, ...

Inn(''G'') (its

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 ...

by its center) is simple (and it follows Inn(''G'') must be non-abelian simple, as inner automorphism groups are never non-trivial cyclic). All non-abelian simple groups are quasisimple. The subnormal quasisimple subgroups of a group control the structure of a finite insoluble group in much the same way as the minimal

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 ...

s of a finite

soluble 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 terminat ...

do, and so are given a name,

component Component may refer to: In engineering, science, and technology Generic systems *System components, an entity with discrete structure, such as an assembly or software module, within a system considered at a particular level of analysis * Lumped e ...

. The subgroup generated by the subnormal quasisimple subgroups is called the layer, and along with the minimal normal soluble subgroups generates a subgroup called the generalized Fitting subgroup. The quasisimple groups are often studied alongside the simple groups and groups related to their

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 ...

s, the

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 ...

s. The

representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...

of the quasisimple groups is nearly identical to the projective representation theory of the simple groups.

Examples

The covering groups of the alternating groups are quasisimple but not simple, for

n \geq 5.

Examples

See also

References

External links

Notes