In mathematics, a quasisimple group (also known as a covering group) is a

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that is a

perfect Perfect commonly refers to: * Perfection, completeness, excellence * Perfect (grammar), a grammatical category in some languages Perfect may also refer to: Film * Perfect (1985 film), ''Perfect'' (1985 film), a romantic drama * Perfect (2018 f ...

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 da ...

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

short exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the conte ...

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

such that

E =

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/math>, where

Z(E)

denotes 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 ''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 I. Martin "Marty" Isaacs is a group theorist and representation theorist and professor emeritus of mathematics at the University of Wisconsin–Madison. He currently lives in Berkeley, California and is an occasional participant on MathOve ...

, ''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 simple operations from within the group i ...

Inn(''G'') (its

quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...

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 Subnormal may refer to: *Subnormal body temperature, a common term for hypothermia * Subnormal operator, a type of operator in operator theory in mathematics *Subnormal number, another name for a denormal number in floating point arithmetic *Subnorm ...

quasisimple subgroups of a group control the structure of a finite

insoluble 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 terminate ...

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 G ...

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,

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. 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 In mathematics, especially in the area of algebra known as group theory, the Fitting subgroup ''F'' of a finite group ''G'', named after Hans Fitting, is the unique largest normal nilpotent subgroup of ''G''. Intuitively, it represents the small ...

. 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 th ...

s, the

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

s. The

representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...

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