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 ...

, in the field of

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...

, a component of a

finite Finite may refer to: * 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 marked for person and/or tense or aspect * "Finite", a song by Sara Gr ...

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 a quasisimple

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''. In n ...

. Any two distinct components commute. The product of all the components is the layer of the group. For finite abelian (or

nilpotent In mathematics, an element x of a ring (mathematics), ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term, along with its sister Idempotent (ring theory), idem ...

) 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 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 ...

has even order, it is normal in the centralizer of every

involution Involution may refer to: Mathematics * Involution (mathematics), a function that is its own inverse * Involution algebra, a *-algebra: a type of algebraic structure * Involute, a construction in the differential geometry of curves * Exponentiati ...

centralizing it, and it commutes with none of its conjugates. This concept is used in the

classification of finite simple groups In mathematics, the classification of finite simple groups (popularly called the enormous theorem) is a result of group theory stating that every List of finite simple groups, finite simple group is either cyclic group, cyclic, or alternating gro ...

, 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 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 .

References

* * Group theory Subgroup properties {{group-theory-stub