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 paranormal subgroup is 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 ...
such that the subgroup
generated by it and any
conjugate of it, is also generated by it and a conjugate of it ''within'' that subgroup.
In symbols,
is paranormal in
if given any
in
, the subgroup
generated by
and
is also equal to
. Equivalently, a subgroup is paranormal if its weak closure and normal closure coincide in all intermediate subgroups.
Here are some facts relating paranormality to other subgroup properties:
* 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, .
...
, and hence, every
normal subgroup and every
abnormal subgroup, is paranormal.
* Every paranormal subgroup is a
polynormal subgroup
In mathematics, in the field of group theory, a subgroup of a 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 i ...
.
* In finite solvable groups, every polynormal subgroup is paranormal.
External links
{{cite book, authorlink1=William Kantor, last1=Kantor, first1=William M., last2=Martino, first2=Lino Di, title=Groups of Lie Type and Their Geometries, date=12 January 1995, publisher=Cambridge University Press, isbn=9780521467902, pages=257–259, url=https://books.google.com/books?id=iXXAZ3dmkNwC&dq=Paranormal+subgroup&pg=PA258
Subgroup properties