In mathematics, 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 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 ...

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 a

quasisimple 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/ ...

subnormal subgroup. Any two distinct components

commute Commute, commutation or commutative may refer to: * Commuting, the process of travelling between a place of residence and a place of work Mathematics * Commutative property, a property of a mathematical operation whose result is insensitive to th ...

. The product of all the components is the

layer Layer or layered may refer to: Arts, entertainment, and media * ''Layers'' (Kungs album) * ''Layers'' (Les McCann album) * ''Layers'' (Royce da 5'9" album) *"Layers", the title track of Royce da 5'9"'s sixth studio album * Layer, a female Maveri ...

of the group. For finite abelian (or

nilpotent In mathematics, an element x of a 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 was introduced by Benjamin Peirce in the context of his work on the cla ...

) 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 of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', ...

has even order, it is

normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...

in the centralizer of every

involution Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...

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 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 i ...

, 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

* *{{Citation, last1=Aschbacher, first1=Michael, author1-link=Michael Aschbacher, last2=Seitz, first2=Gary M., title=On groups with a standard component of known type, journal=Osaka J. Math., volume=13, year=1976, pages=439–482, issue=3 Group theory Subgroup properties