In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
''H'' of a given
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 ...
''G'' is a subnormal subgroup of ''G'' if there is a finite chain of subgroups of the group, each one
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 next, beginning at ''H'' and ending at ''G''.
In notation,
is
-subnormal in
if there are subgroups
:
of
such that
is normal in
for each
.
A subnormal subgroup is a subgroup that is
-subnormal for some positive integer
.
Some facts about subnormal subgroups:
* A 1-subnormal subgroup is a proper normal subgroup (and vice versa).
* A
finitely generated group
In algebra, a finitely generated group is a group ''G'' that has some finite generating set ''S'' so that every element of ''G'' can be written as the combination (under the group operation) of finitely many elements of ''S'' and of inverses of s ...
is
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 class ...
if and only if each of its subgroups is subnormal.
* Every
quasinormal subgroup __NOTOC__
In mathematics, in the field of group theory, a quasinormal subgroup, or permutable subgroup, is a subgroup of a group that commutes (permutes) with every other subgroup with respect to the product of subgroups. The term ''quasinormal su ...
, and, more generally, every
conjugate-permutable subgroup, of a finite group is subnormal.
* Every
pronormal subgroup In mathematics, especially in the field of group theory, a pronormal subgroup is a subgroup that is embedded in a nice way. Pronormality is a simultaneous generalization of both normal subgroups and abnormal subgroups such as Sylow subgroups, .
...
that is also subnormal, is normal. In particular, a
Sylow subgroup
In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixed ...
is subnormal if and only if it is normal.
* Every 2-subnormal subgroup is a conjugate-permutable subgroup.
The property of subnormality is
transitive, that is, a subnormal subgroup of a subnormal
subgroup is subnormal. The relation of subnormality can be defined as the
transitive closure
In mathematics, the transitive closure of a binary relation on a set is the smallest relation on that contains and is transitive. For finite sets, "smallest" can be taken in its usual sense, of having the fewest related pairs; for infinite s ...
of the relation of normality.
If every subnormal subgroup of ''G'' is normal in ''G'', then ''G'' is called a
T-group.
See also
*
Characteristic subgroup
In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphism ...
*
Normal core In group theory, a branch of mathematics, a core is any of certain special normal subgroups of a group. The two most common types are the normal core of a subgroup and the ''p''-core of a group.
The normal core Definition
For a group ''G'', the nor ...
*
Normal closure
*
Ascendant subgroup
In mathematics, in the field of group theory, a subgroup of a group is said to be ascendant if there is an ascending series starting from the subgroup and ending at the group, such that every term in the series is a normal subgroup of its successor ...
*
Descendant subgroup
In mathematics, in the field of group theory, a subgroup of a group is said to be descendant if there is a descending series starting from the subgroup and ending at the group, such that every term in the series is a normal subgroup of its predec ...
*
Serial subgroup
In the mathematical field of group theory, a subgroup ''H'' of a given group ''G'' is a serial subgroup of ''G'' if there is a chain ''C'' of subgroups of ''G'' extending from ''H'' to ''G'' such that for consecutive subgroups ''X'' and ''Y'' in ' ...
References
*
* {{citation, first1=Adolfo, last1=Ballester-Bolinches, first2=Ramon, last2=Esteban-Romero, first3=Mohamed, last3=Asaad, title=Products of Finite Groups, year=2010, publisher=Walter de Gruyter, isbn=978-3-11-022061-2
Subgroup properties