In
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
subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation ...
''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 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 o ...
is
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 ...
if and only if each of its subgroups is subnormal.
* Every
quasinormal subgroup, and, more generally, every
conjugate-permutable subgroup, of a finite group is subnormal.
* Every
pronormal subgroup that is also subnormal, is normal. In particular, a
Sylow subgroup 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 homogeneous binary relation on a set (mathematics), set is the smallest Relation (mathematics), relation on that contains and is Transitive relation, transitive. For finite sets, "smallest" can be ...
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
*
Normal core
*
Normal closure
*
Ascendant subgroup
*
Descendant subgroup
*
Serial subgroup
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