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
Traditionally, a finite verb (from la, fīnītus, past partici ...
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 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 ...
. Any two distinct components
commute. The product of all the components is the
layer
Layer or layered may refer to:
Arts, entertainment, and media
*Layers (Kungs album), ''Layers'' (Kungs album)
*Layers (Les McCann album), ''Layers'' (Les McCann album)
*Layers (Royce da 5'9" album), ''Layers'' (Royce da 5'9" album)
*"Layers", the ...
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 cl ...
) 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 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 input ...
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 els ...
, 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