In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, in the field of group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

, a component of a finite

Finite group

In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...

 group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

 is a quasisimple

Quasisimple group

In mathematics, a quasisimple group is a group that is a perfect central extension E of a simple group S...

 subnormal subgroup

Subnormal subgroup

In mathematics, in the field of group theory, a subgroup H of a given group G is a subnormal subgroup of G if there is a finite chain of subgroups of the group, each one normal in the next, beginning at H and ending at G....

. Any two distinct components commute

Commutativity

In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...

. The product of all the components is the layer of the group.

For finite abelian

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

 (or nilpotent

Nilpotent group

In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...

) groups, p-component is used in a different sense to mean the Sylow p-subgroup, so the abelian group is the product of its p-components for primes p. These are not components in the sense above, as abelian groups are not quasisimple.

A quasisimple subgroup of a finite group is called a standard component if its centralizer has even order, it is normal

Normal subgroup

In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....

 in the centralizer of every involution centralizing it, and it commutes with none of its conjugates. This concept is used in the classification of finite simple groups

Classification of finite simple groups

In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic...

, for instance, by showing that under mild restrictions on the standard component one of the following always holds:

• a standard component is normal (so a component as above),
• the whole group has a nontrivial solvable
  Solvable group
  In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
  
  normal subgroup,
• the subgroup generated by the conjugates of the standard component is on a short list,
• or the standard component is a previously unknown quasisimple group .